Here are some possible words:
megalith
megillah
midmonth
mishmash
mishmosh
mismatch
monolith
moonfish
Cash is a 4 letter word that starts with c and ends with h.
abolishaccomplishadmonishafterbirthamateurishaftermathanguishapproacharchashastonishattachautographazimuth
Hosanna
'Haiku' (hye-KOO)
Well, one that does is "humor".
hush
Its HANDSOME
Heart
Birth
handsomenessharmlessnesshelplessnesshomelessnesshopelessnesshorribleness
Hostile
Firehose