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I'm marking a students homework and have come across the phrase "the increased chances of the local extinction for this specie" and I'm having trouble with "the local extinction" as I think it sounds more natural to say "a local extinction of this species" but I can't explain why. This is talking about a specific event so I want to say it should be 'the' but my heart says indefinite. Any help? Cheers

Maybe with "local" the indefinite article is possible since you can indeed have multiple local extinctions as opposed to only one global extinction. But depending on your viewpoint (geographically - are you looking at THIS locality or many?) the definite article might also be possible. Hard to say without more context.

You can't put indefinite article before "extinction", because it is uncountable noun. I would omit the article. If we want to talk more specifically and to differentiate this extinction from another, we should add a noun, for example "instance of extinction", though in my opinion it looks a bit clumsy.
It is the same as with "evidence", "information" or "advice". We have to use "piece of" if we want to make it plural.

Your idea to use the indefinite article could be appropriate as it can have the function to be a form of one e.g. A local extinction, refers to one extinction.