Subjunctive mood

"I hope that he stays here"

I understand. "Ihope he stays here" is more correct of the two. But can I say also "I hope that he stay here"? Would that be correct as well?

no, that would be bad grammar...

We use the subjunctive to talk about events that are impossible or unlikely. You could say "I wish he would stay here" if you know or expect he will not stay. To express a concrete hope that is possible we use the present "I hope he stays here".

No. Incorrect. See above.