the structure many + a + noun is used to emphasise that there are a lot of people or things involved in something.
ex1. many a mother tries to act out her unrealised dreams through her daughter.( Lots of mothers...)
ex2. i have spent many a morning with my wife collecting seashells along the beaches of Samos. (I have spent lots of mornings...)
"I spent many lazy Sunday afternoons that way" is ok as well.
The only difference I can see between the two is that the first one is more emphatic.