Depending on how wide how wide these smaller side tunnels are, I would imagine even the power armour units would be too large to effectively fight in these environments. Assuming that the side tunnels are bored out purposely for humans to pass through, the ceiling is probably going to be 7 or 8 feet high at best, meaning the power armour units are not going to be able to stand upright. Again, the width of these tunnels are probably suited such that people can pass 2-3 abreast. Considering that the power armour units are going to be significantly wider than the average human, plus their weapons (which I imagine are going to be quite a bit larger than the average infrantry weapon) and miscallaneous peices of kit such as radios, food, water, medical supplies, explosives etc, it's unlikely the power armour units would be able to even turn around in these tunnels easily, let alone fight.
Having a mech fight in the side tunnels is probably going to be like how the tunnel rats had to fight in Vietnam. (Another real world example of people trying to navigate confined spaces with quite alot of cave diving- getting your kit stuck on rocks in these tight spaces can be a death sentence). Imagine a small team of mechs inside one of these passages and for some reason they need to quickly fall back- likely their only option would be to shuffly backwards until they get to an opening where they have the room to turn around.
This is in addition to the lack of fine motor control mechs will have, so trying to navigate particularly narrow cave is going to be a nightmare for those troops.
Power armour troops would probably adopt similar tactics to the US in Afghanistan (instead of clearing out a cave, just use explosives to collapse all the entrances and exits, and go on your merry way while the men inside struggle to dig themselves out).
Your average squad of infantry, armed with a carbines and shotguns, would likely be much better and fighting in these confined spaces, especially considering that if the mechs can't follow them, enemy infantry will flee in to them whenever they are outgunned.
Edited based on comment:
Provided that the power armour troops stay within those car sized tunnels, then I imagine they probably would be well suited. They'd be a big target, and probably slower to react in close quarters than light infantry, but given that much of it will be fighting in low light environments due to smoke and debris from hand grenades, gunfire, damaged lights etc, the more sophisticated optics that I imagine they'd have in their helmets would be very useful- though on the other hand I imagine they'd have significantly narrower field of vision through their helmets, so they'd rely on light infantry for support in close quarters.
Also, considering the amount of richocets from shrapnel and bullets hitting the sides of the tunnels (04:00 onwards in this FBI video is very informative on this), as well as the amount of direct fire directed at them fighting in narrow chokepoints, having your point man as a walking shield would definitely improve the life expectancy of any light infantry following.
Depending on the amount of armoured troops that this military has, they may struggle to field enough power armour troops if there are a lot of tunnels to cover (and the power armour troops would probably be overfatigued from constant deployments in any longer term wars), though they'd definitely be a valuable resource.