Learn Turkish: Neyse, hadi gel.

Neyse is actually a combination of ne (what) and ise (if it is). Literally it means something like "whatever it is." In everyday Turkish, it is the standard way to say "anyway," "well," or "whatever" to change the subject or wrap up a thought.

What is the exact purpose of hadi here?

Hadi (often also spelled haydi) is a very common word used to encourage someone to do something. It translates best to "come on," "hurry up," or "let's." When you put it in front of a command like gel, it makes the request feel more like an enthusiastic encouragement rather than a strict order.

Why is the verb just gel without any endings?

In Turkish, giving a direct command to one person informally (a "you" or sen command) is extremely simple: you just use the root of the verb! The dictionary form is gelmek (to come). Drop the -mek, and you are left with gel ("come!").

Could I say this to anyone, or is it too informal?

This sentence is quite informal. Because it uses the bare verb root gel and the casual hadi, you should only use it with friends, family, or people younger than you. For someone you need to be polite with, or for a group of people, you would say Hadi gelin.