First off, prepositions are tricky words, and it's hard to talk about all the ways they can be used.
That said...
at
We can use at night to mean during the nighttime:
I was there at night.
(notice how I did not say "at the night," which is what you wrote in your question).
in
We can use in the night to mean during the nighttime or at various times during the night:
Hyenas prowl in the night. (at night would work, too)
into
We use into the night to say that something continues for a long time after sunset:
The partygoers reveled late into the night.
on
You are correct, we don't usually say on night or on the night. However, if we expand the phrase, and stipulate which nights we are talking about, then "on (the) nights when..." is a relatively common construct:
The band liked crashing at our place on nights we played at Live Bay.
I try to go to bed early on Sunday nights.
On the nights when we hear the ice cream truck, we'll go outside and buy ice cream.
Now, let's combine some of these together:
We usually lock the front door at night. However, our roommate Micheal is an actor, and he sometimes works late into the night. On the nights Micheal hasn't come home before midnight, we leave the front door unlocked, in case he has forgotten his key.
None of them seem completely natural ("compensation" seems a bit formal), but in that specific context, I'd actually suggest "for compensation", because it mirrors the question:
Could you work on this for free?
No, but I can work on that for compensation.
or even better,
No, but I can work on that for pay.
Best Answer
Both "on" and "over" are acceptable prepositions to use with "influence"; however in this context of a scientific experiment, "on" is probably the more idiomatic choice. In most contexts, "on" implies a variable or unknown degree of influence. Example:
"Over" implies a greater degree of influence, or influence over a wider range. For example:
Here we could use "on", but it would change the meaning, as the writer wants to say that the senators implied the President did improperly influence the investigation.