Perhaps something like this:
ReplacePart[a, #1 -> Append[a[[#1]], #2]& @@@ b]
If data are large and speedup is needed you might want to try Join
ReplacePart[a, #1 -> Join[a[[#1]], {#2}] & @@@ b]
In case you want value of a to be the new list with added elements, you whether can use AppendTo:
ReplacePart[a, #1 -> AppendTo[a[[#1]], #2] & @@@ b]
or simply reassign:
a = ReplacePart[a, #1 -> Append[a[[#1]], #2] & @@@ b]
