I use Filmora to add just a basic filter and audio to my drone shot but it doesn't really have much to offer as far as cleaning up footage. Is there a free or cheap software that you use to clean up your videos?

Long ago, I used to use "Magix Audio Cleaner" app. Did the job good.

I'm sorry to tell you, but all the softwares that can do a good video cleaning is expensive.
if you want better understand  what that softwares can do, you can search the plugin: Neat Video

Do you mean audio or noise?

Montfrooij Posted at 2017-12-9 10:26
Do you mean audio or noise?

The white grain that comes from night video

Astro832 Posted at 2017-12-9 11:18
The white grain that comes from night video

I thought so.
I never removed that, but that is because I don't have the funds for a good plugin that will do the trick.
The one in Sony Vegas is not so good.
I have seen nice results from Neat video, but that is a bit expensive.