Iran’s oil is sanctioned hence not on public market. Does it really have much influence?

China buys Iranian oil, if they’ll start to but oil from non-sanctioned countries it will push prices up. But the biggest reason for prices to go up is the risk that Iran will attack tankers in the strait of Hormuz or oil infrastructure on Arabian peninsula.

Yes, because if it stops flowing, demand on the public market will increase, and prices will rise.