This has come up in plenty of literature I've seen. The usual explanations come down to either it's too expensive to do so, some other force prevents them, or they just can't be bothered to. Here's a quick list of the main options, in no particular order.
God's just don't work that way.
I doubt this is the option you will want to go with, but it's possible divinity just doesn't work in such a way. Maybe your god doesn't have a physical body, and works mostly by inspiring his disciples to do something. He just doesn't have the ability to show up or do overt miracles because deities are more limited then that.
Your god wants faith, confirmation of his existence removes the need for faith
Your god could prove himself, but he wants his priest to work out of faith without proof. If your god proves himself to the unbelievers he also proves himself to his believers, and in so doing removes the ability of them to have faith in him without proof. There are a few variants of this option.
Gods are powered by (or otherwise benefit from) faith, the kind of faith that only is possible if one believes in something without absolute proof.
This is a test of his followers, so they can prove they are wiling to support him even in hardship or without faith (which is kind of a jerk move, but hey the majority of gods in religions kind of are jerk).
It's a temporary test, he will get around to helping eventually
A slightly less jerk version of the last option. He is temporarily refusing to prove himself to test his faithful. He want's to see if they will past his test by continuing to have faith, stand up to hardship, etc. Only after they prove themselves will he show himself. So maybe he doesn't do a miracle on demand and his faithful are captured, but then he does a different miracle to help them all escape once they refuse to refute him even in that situation.
For a biblical comparison look at when god asked Abraham to sacrifice his son, then told Abraham he passed and didn't actually need to do it when Abraham showed he was willing to do so.
Your god doesn't want his faithful to be dependent on, or make demands of, him
If he shows up now and does a miracle for you why wouldn't he the next time you're in trouble? What happens when a priest offers to have his god preform a miracle to earn a room for the night? A believer gets in a drunken bar fight and offers up all his money if his god doesn't prove his existence right now?
In short this is the slippery slope argument. If his faithful can demand his appearance because it would be convenient for them they may become dependent on that. He wants them to be able to function without his constant intervention and daily proof of his existence, and so they need to learn to function without on-demand miracles; even if the lack of them is going to be a significant hardship.
God helps those who help themselves
A very close variant of the last idea here. Your god only intercedes to help people after they have put in will effort and done what they can by themselves, so they don't get lazy and dependent on him. Thus he won't do his miracle yet because he needs his people to first 'help themselves.' In this case that might mean they have to fight in this competition on their own, at least at first, and prove themselves before he will come in and help out at the end.
You have angered me and the LORD is vengeful and strong in wrath
Maybe your god doesn't help because you have angered him and no longer deserve his aid. Perhaps he didn't want this competition, or he is angered they make demands on him to prove himself. Maybe they have done something entirely unrelated to fail him but their punishment is his lack of help here. The point is his lack of aid is because of their short comings.
You might recognize the second half of this section title is actually a biblical quote, and for good reason. There are a multitude of examples in the bible where God's chosen people suffer defeats, hardships, and humiliations. From being enslaved in Egypt to having to wander a desert for 40 years to countless military failures. In practically ever case the reason the bible gives for God not intervening to aid his people is that they had done something to anger him and prevent him from coming to their aid; in fact some of the hardships were actually intentionally inflicted by God as punishment. There are plenty of examples to draw on for why your god might end up unwilling to help his faithful.
The nature of divinity prevents an overt show of power
Even god's aren't omnipotent and are limited in what they can do. I've actually seen many variants of this idea, but they tend to be similar enough I won't list them all.
The most common version though is that there is another even higher being, entity, or simple nature of the world and divinity that sets limits on what gods can do. This limit prevents the god from doing the sort of overt demonstration.
For instance maybe miracles have to have a plausible explanation. If a god does something so overtly impossible as to have no other explanation he will incur the wrath of a greater entity or trigger some other terrible consequence. Perhaps anyone that sees an overt miracle without a plausible explanation will somehow be driven made with the revelation. Maybe witnesses to such miracles will simply be immediately killed by the greater power that doesn't want humans to see undeniably proof that deity can work outside the realm of nature.
That's not how the god of X works
Many gods in fiction are gods of certain ideas or concepts, and those concepts don't only define them but also may constrain them. The god of fire is not going to, and might not able to, do a miracle involving creating a giant ice sculpture, because he isn't the god of ice. Their godhood, and the concept they are tied to, thus set rules and limits on how they manifest their powers which may in turn prevent them from manifesting here.
Here we can piggyback a bit off of past ideas for inspiration on what your god's nature and thus limits might be preventing his manifesting here. If your god is the god of faith he may be dependent on his people acting on faith instead of demonstrations for instance. If he is the god of humanity and connections he may work through inspiring his people, suggesting they help each other, moving them in the right place to do the most good etc; but in turn since he works through people he doesn't do great miracles. Alternatively perhaps he is the god of luck/timing/coincidences who inspires his people to keep an eye for the right time to move and his miracles work by slightly nudging odds to make coincidences work in his folks favor. In this case he may be helping his people all the time, but because he is only nudging odds to make things work out better for his people even when he works a miracle it will look like a coincidence, and as such won't be very convincing as a miracle.
I could go on and on, but since I suspect this isn't he path you will take so I won't waste my effort. The point is it's easy to make the nature of his particular power and providence one that doesn't lend itself to fancy overt miracles, even as other gods with different powers can do them.
It's part of the goods deal/bet
The two gods in this dispute have an agreement that they will allow their faithful to solve this situation without their direct involvement; thus neither side can actively get involved. There are many reasons for this.
There might be some convention or pact between all gods which sets rules for how they interact in the world, and some of those rules include allowing disputes between two set of faithful to happen without direct intervention.
Close variant of the last option: Direct intervention in faithful disputes is the nuclear option. That is to say anyone could do it, and if they did it would be highly effective. Unfortunately as soon as they do they open up the door for everyone else to do the same. A world where gods are constantly actively participating in fights between faithful is considered undesirable by the gods, either because it's too costly for the gods or because it would lead to so much devastation to humans. Thus there is a sort of unwritten agreement no one will intervene in such disputes because they don't want to force the other gods to do the same in the future.
The more established and powerful deity has been convinced to not get involved via plea from the new god, deal with other gods, general good sportsmanship, or some other contrivance to not act directly so long as the other god doesn't. The new god considers not proving himself to be an acceptable sacrifice to get the other deity to agree to not get involved since he knows he would lose in a competition in which the two were directly involved. Thus he stays hands off so the other one will as well.
It's a trial by champion, using their believers to solve their dispute so they don't have to. God vs God battles are bloody and everyone looses. Thus the gods usually allow their faithful to handle disputes in their place. They have set laws and rules for how such disputes can be handled, which allow certain types of aid to their followers, but forbids others. The rules of such trial/disputes prevent the sort of overt intervention you suggested.
It's all part of the bet. Remember when I said many gods are jerks? well that's going to be a reoccurring theme in these answers. Perhaps your gods aren't treating this as a life and death competition, but instead as a fun parlor bet. They want to see what's happening and have made a good natured bet on the results. Now their let those petty human things, which they consider barely more then ants and don't really care if their hurt, play out their dispute to see what happens and which god needs to buy the other an ambrosia beer.
The other god is preventing him
Nice simple one, the more powerful god is powerful enough to get in the way of the weaker one. The weaker one can't use his power as long as the stronger one is blocking it.
He hasn't yet risen to his proper power as a god
Your god is weaker and lesser known, there is a reason for that. He had been locked away in some prison in the past, or injured, or is new and still developing into his godly strength. Whatever the cause he is currently handicapped and limited, and his people know that. They hope to help him grow to his full power, and in so doing gain his blessing and future support for their aid. However, in the meantime it's known he is quite limited and may not yet be able to pull of an impressive miracle.
He deems it too expensive
Gods may not be all powerful. They can do miracles, but doing them isn't easy. It requires some expense of effort, maybe they gain power from faith but then expend it when doing wonderous things for instance. The point is there is a cost to do a miracle and it's non trivial.
The god would like to do something here, but he is still weak and has only so much strength. With limited power to work with your god has decided that his resources are better spent in some other way and he just can't afford the expense of doing a miracle here. He doesn't like doing that, but deems it the wisest use of his resources either for himself or for his followers if he is more altruistic.
Here I'd point you to an old game Black & White where you play as a god. You often had exactly this tactical decision. Miracles cost faith, you can gain faith by doing a miracle in front of people, but not enough faith to cover the cost of the miracle. You need to pick and choose where to perform your miracles to get the most bang for your buck if you want to beat your rival god. Sometimes that means choosing not to do a miracle to conserve power.
This option also can be combined with other options, such as the rival god blocking him, to help explain why it's so costly to do a miracle right now.
It's all going according to plan
As an all knowing deity he can make plans beyond that of mortal man. With this brilliant knowledge he has made some great plan that involves his faithful loosing this competition. Given the neigh-omnipotence of gods you can get away with any level of convoluted reasons for why the god was convinced not helping his followers would better further his long term plans.
It has been prophcied
A prophecy from another god or power suggests that this fight will be lost anyways, or that some greater evil will happen if the god gets involved, so the god chooses to stay away.
On a particularly jerky god pessimistic version the god made a prophecy to his people in the past and realized it won't come true if he helps here. So he lets his people fail here just so he can brag that his prophecies came true. Basically this is the godly version of a politician being more concerned about looking good politically then making good laws that will benefit their constituents.
We're sorry but your god is busy on the other line. Please keep prostrating yourself and he will get back to you as soon as he can. Please know your prayer is important to us.
Plenty of works I've seemed have non omnipotent gods, who for all their power can only do so many things at once. In this case they may be distracted or too busy to help their faithful at this moment because another situation has come up.
Your god may be helping faithful in some other part of the world right now, in a more complicated situation that simply needs their intervention more.
Your god may have been intentionally distracted by another god to keep him from being available to help you, or a god may be using some divine power to help hide you from your gods sight so he doesn't notice you need him.
Your god is lazy and was too busy playing in the Divine Games, getting drunk on ambrosia, or sleeping in to bother helping you right now. Have I mentioned yet that many gods were kind of jerks?