short of adding phonetic pronunciations of words into the NVDA default or voice dictionaries - something I frequently do becuase, living in Wales, there are lots of words that synth voices don't know how to pronounce, I am unaware of any way to adjust what a synth voice says. The other thing that complicates matters is that the letter h is often effectively silent due to its usual mid-word pairing with other letters that provide the actual sound .. ch, th, sh ...
But, I tried using the Mark voice and cannot replicate your missing h sounds whether they appear at the start of the word ... all h's in the following sentence sound correct when I use Mark at speed 75 on my Asus laptop:
hello all this is a hotel in Honolulu and I was very happy here this Halloween full of horrible ghosts
And I also tried to think of words where an h appears mid-word with a sound of its own and came up with one from my chemistry student days, anhydrous, and it pronounced that fine too!
Sorry that's no hlpe ... have you tried using a USB soundcard to see if that changes the pronunciation? I presume other voices manage the h sound that Mark is missing?