Nested Map and Apply

One option is to separate the slots by using an explicit Function for the second argument

Map[Function[arg, Apply[{h[arg, ##]} &, {a, b}]], {1, 2}]

Regarding your updated question. The approach is the same

Map[Function[arg, Apply[{h1[arg, ##], h2[arg, ##]} &, 
  {RandomReal[], RandomReal[]}]], {1, 2}]

Simply you could use:

Thread @ h[{1, 2}, a, b]

{h[1, a, b], h[2, a, b]}

If you can demonstrate how that fails in your application I will give other methods.

---

It was suggested that I use Sequence @@ {a, b} so as to keep {a, b} in the given form. I did not, because I was not clear as to the expected input format and because I felt that it would obscure the syntax.

Taking a guess as to your desired syntax, you might use:

f[head_][{q__}, {r__}] := Thread @ Unevaluated @ head[{q}, r]

f[h][{1, 2}, {a, b}]

{h[1, a, b], h[2, a, b]}

Unevaluated is needed for cases such as:

f[Print][{1, 2}, {a, b}];

1ab

2ab

---

Based on your updated question perhaps you want:

Through[{h1, h2} @@ RandomReal[1, 2] ~Prepend~ #] & /@ {1, 2}

Or

Through[{h1, h2}[#, Sequence @@ RandomReal[1, 2]]] & /@ {1, 2}

But if this is really representative of your usage there is probably a faster way.

Version 11.3 introduced Curry which can be used to good effect here:

Map[Apply[Curry[{h[##]} &, {3, 1, 2}], {a, b}][#] &, {1, 2}]

(* {{h[1, a, b]}, {h[2, a, b]}} *)

Map[Apply[Curry[{h1[##], h2[##]} &, 3][#], {RandomReal[], RandomReal[]}] &, {1, 2}]

(* {{h1[1, 0.373787, 0.928873], h2[1, 0.373787, 0.928873]},
    {h1[2, 0.271899, 0.0866199], h2[2, 0.271899, 0.0866199]}} *)