How do the option writers influence share price? I have heard about short covering but it happens only in FNO how can it help the stock price to up since the shorts are built in FNO and its derived from cash not the other way round?
For the most part, when a trader sells an option, it has no effect on the underlying because it's a trade between two counter parties on the option market.
There may be a mild effect when the options are exercised if the assigned party does not own (or is short) the underlying but this has no significant effect in liquid stocks.
The exception to this would be a gamma squeeze. When a market maker sells a call, he buys shares equal to the option's delta to remain delta neutral. If there is massive call buying and there's a short squeeze occurring, the market maker's share purchases further fuel share price increase. The classic example of this occurred last year when Gamestop rose from about $20 to $513 in two weeks.