Replacing multiplication of matrix elements by application of these elements as functions

I want to realize the following idea in Mathematica.

I've got a matrix

{{a,b},{c,d}}

which is multiplied to a vector {h,k} leading to

{{a h + b k}, {c h + d k}}.

Imagine now that a is an operator and I want to apply it to h, instead of multiplying. Primitive substitution {a h -> a@h} helps, but it is not quite a good approach while it's not working in more sophisticated cases.

Thank you!

EDIT1: The problem is solved partially, all comments are very useful. But still I am a little bit stacked, so I'm posting the update trying to explain my exact problem.

The problem is following. I want to construct the matrix

{{a[x], b},{c, d}}

where a[x] is an operator (function) and b,c and d are arbitrary expressions (which are symbolic in general). After applying the operation

{{a[x], b},{c, d}}.{{h}, {k}}

I want to obtain

{{a[h] + b k}, {c h + d k}}

I want this operation to work not only for numbers and functions as it was proposed in answers below but with arbitrary symbolic expressions. I mean I want Mathematica to understand that if x and p are not functions but just variables, then x*p means multiplication, otherwise it means x[p].

Moreover I want this operation to work in more general cases, e.g.

{{a[x], b},{c, d}}.M.Transpose[{{a[x], b},{c, d}}],

where M is an arbitrary matrix.

I would be very grateful for any ideas.

Answer

Picking up on Marius tip on Inner in the comments:

Inner[Apply[#1, {#2}] &, {{a, b}, {c, d}}, {h, k}]

And @ciao offered a better version in comments:

Inner[#1[#2] &, {{a, b}, {c, d}}, {h, k}]

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