Can we use "a" after "many"? What does it mean, then?
Example:
Every living thing (and perhaps many a dead one as well) pays heed to that call. source
Yes you can use 'a/an' after many.
This is a more formal or old-fashioned expression I suppose.
Like the adjective - (She was depressed for many months) and
pronoun form (Some workers will be willing to work on sunday, but many will not.) of many, Many a/an is used to indicate a large number of something.
However it takes a singular noun.
Some Examples:
She remained hidden from the public for many a year. [for many years]
I've been to Paris many a time. [= many times]
Many a politician has promised to make changes. [Politician and has are singular].