Is it correct to say hang in there?

"Hang in" means to endure. It's often an exhortation, as in "Hang in there, you'll be alright", but it can be descriptive as well: "Muhammed Ali was winning, but Norton was hanging in there." "Hang on" can also be an exhortation to endure, but it can also mean, "please wait": "Hang on for a moment while I check".

Which is correct hang in there or hang on there?

In the first example, the verb is just 'to hang'. In is a preposition and is used because it has the sense of in the game or in play. ... People do say, "Hang on there", or "Hang on a minute", but here the verb is the phrasal verb 'to hang on'.