1st, what do these could do gently in the night in that zone? ;) Well fun aside, I thing they do drop drops of water.
So... even more interesting, what kind of desert this may be? And whats below? Sand Deserts, while the ones, that people think of if they hear the word "desert", just made a small part of our worlds deserts.
So it's likely that's a rocky wasteland. It could be an ice-desert too.
What does desert mean? If I recall right, its a place with fairly few natural appearances of water. Long time ago my teacher try to teach me what a desert make a desert, and all that's left is a vague impression of an area that contains pretty few water and does encounter even less atmospheric fallout. An arid environment. Guess I was way to busy daydreaming in school.
What does this tell us? The ground, while he might soak up some of this, will be likely to refuse soaking up even more thanks to millenia of dry... ehm... whats the word... compression?
Never mind - your Rain would gather together and flow away without having much of an effect. Temporary Rivers in deserts caused by this seems to make more people drown in a desert than actually die of thirst. In Australia you should immediately leave your piece of desert, if you see heavy rain far away, because few time later (hours?) you will be greeting by a muddy flood. Well, just follow the kangaroos, they know how to survive this.
Back to topic - when your Rain-Wizard aims for a valley, he might create a lake. When every night as much water is refilled as evaporated during day-time, you might get a real oasis over time. But once the rain overnight cuts off, it's likely to dry out again. And your plants will die.
And for the effects... if you create this rain out of nothing, you will encounter a rise in sea-level in a few thousand years. But when you just soak this from air humidity, ... hard to say. Because its an arid environment, there will not much humidity you could catch. Maybe collect this over day.
But anyway, 30 Miles is a pretty cute size compared to a planet wide climate. And at the end, that water will find its way back to the circle of water, so for me I would not except much changes.
Okay, for the sake of story-telling, you could go with the usual cloud movement and look whats there. A rainforest would barely notice your two hours of gently rain, but an arid area with inhabitants that need these drops you claim for your desert may find themselves a bit shorter of water supply than usual. Which may be pretty unpleasureful.
Maybe a suggestion... even if the surface of a desert is dry as... a desert, you may find a pretty river if you go in the other direction: not for clouds, but for the underground.
Down there its cool and the water wont evaporate.
Another think - when your desert is surrounded by mountains, you will get your water using a... darn... there a kind of well, that looks for aquifer of mountains. Because the part of this located in the mountain is much higher than the one below your desert, all you need to do is drill a hole and let physics move the water up.
-EDIT- Ah, an Artesian Well, thanks Layna!-/EDIT-
A last word: if you don't use the lake-variant and make sure the ground is solid, your water will fall down through the cavernous ground and join the aquifer. If you just rain at a random area, it will most likely flow away. Sand and Stone and Wastelands do not own ground that can store water at their own. You need ... topsoil (?) that will appear though many years of plants claiming their place. MANY years.