Nested Map and Apply

There are some other questions on this topic but I could not get an answer from reading them. What I want to do is use Apply on some of the arguments of a function, h, and then Map on another argument. Here is what I attempted:

Map[Apply[{h[#, ##]} &, {a, b}] &, {1, 2}]
(* ===>{{h[a, a, b]}, {h[a, a, b]}} *)

The elements I want to Map over never get used. But this is not what I want. I want just

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

I could use Table instead of Map but it's slow (slower than just using Apply twice) and I was hoping Map would be faster.

I understand that Apply is using both # and ## but I'm not sure what syntax is correct to force the first Slot to be used by Map instead of Apply.

EDIT: This is more like what I actually want to do:

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

So I want output as

{{h1[1, a1,b1], h2[1, a1,b1]},{h1[2, a2,b2], h2[2, a2,b2]}}

where I a's and b's are the random numbers. So to get this, I think the order of Apply and Map is important.

Answer

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}]