There are a lot of words that end in h!:
with
filth
saith
youth
itch
pitch
ditch
wealth
And if names count,
Joseph
Josh
etc. etc!
Vanquish is an eight letter word. Vermouth is another word beginning with v and ending with h.
Some five letter words ending in H are:batchbeachbelchbirchbirthblushbrashbrushbunchcatchcoachconchcouchcrashcrushdeathearthfetchfifthflashfleshflushforthfreshgnashhatchlatchlunchlurchmarshmatchmorphneathninthnorthpeachperchplushpoochporchreachsixthslushsoothsouthteachtenthtoothtruthyouth
hash
Who are you with
Slashsmashgnashbrashcrashstash.
Hanukkah
astonish
Hearth.
hurrah, huzzah
bath
The answer is botch.It's five letters, ending in H. A botch is an embarrassing mistake.
One is through.