Decompose set into elements?

lst = {{{}, "a"}, {{1}, "a"}, {{2}, "a"}, {{3}, "a"}, {{1, 2}, "a"}, 
   {{1, 3}, "a"}, {{2, 3}, "a"}, {{1, 2, 3}, "a"}}

Thread[{List /@ # /. {} -> {{}}, #2}] & @@@ lst // Column

Alternatively,

Tuples[{List /@ # /. {} -> {{}}, {#2}}] & @@@ lst

same result

I believe this is what you want:

expr /. {
   idxs_ : {__Integer},
   el_
   } :> ({{#}, el} & /@ idxs)

where expr is your expression.