In the spirit of another (if not particularly elegant) way to do it:

MapThread[Append, {test[[All, 1]],Partition[g[#] & /@ Flatten[test[[All, 2]]], 2]}]

In the spirit of another (if not particularly elegant) way to do it:

MapThread[Append, {test[[All, 1]],Partition[g[#] & /@ Flatten[test[[All, 2]]], 2]}]

oops! copied over the wrong version. Try this:

MapThread[Append, {{#} & /@ test[[All, 1]],Partition[g[#] & /@ Flatten[test[[All, 2]]], 2]}] == {{{a1}, {g[a2], g[a3]}}, {{b1}, {g[b2], g[b3]}}, {{c1}, {g[c2], g[c3]}}}

True