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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?

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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.

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    your answers on the site are very useful. on options, what is the short and lucid (in lay ma's terms) way to understand Greeks such as Gama. I can ask a seperate question on the site.
    – puzzled
    Commented Aug 21, 2022 at 15:44
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    Thanks for the kind words. Sorry but I can't help you with your question about Gamma. I've utilized options in a variety of ways for 30+ years. The only Greek that was/is relevant to me is Delta. AFAIC, the rest of them aren't a necessity for the retail trader who employs vanilla strategies (spreads, straddles, strangles, etc.). There are far more important things to be aware of than them.
    – Bob Baerker
    Commented Aug 21, 2022 at 17:53

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