Some words have the letter H as the second letter are:
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ChichShashShethShiahShushThighThothWhich
Chic Ohio
Some four letter words that end in H are:bathwashwishbashhashhushmushpithmythhathpushpath
According to SOWPODS (the combination of Scrabble dictionaries used around the world) there are 2 words with the pattern H-A--H--R. That is, nine letter words with 1st letter H and 3rd letter A and 6th letter H and 9th letter R. In alphabetical order, they are: headchair healthier
I don't know the total number, but some examples are below:shetheacheechobushypushymushypithybatheflashyscathewritheblithesootheraunchyhealthywealthytrashyphotographycalligraphymammographytomographytelepathyempathyapathyantipathysympathybiographybreatheseethescythecrunchycatchyblotchypatchyscratchyfilthyfishystarchymonarchypatriarchymatriarchyworthy