SPINACH
7-letter words ending with H:anguishbeneathbeseechblemishboorishbrutishcherisheyelashfurnishgarnishimpeachmannishmenorahnourishpiggishprudishreddishrematchscratchselfishstealthsunfishtarnishunearthunleashunlatchvarnishwarpath
Warpath, seventh, anguish.
Carvagh
Some five letter words ending in H are:batchbeachbelchbirchbirthblushbrashbrushbunchcatchcoachconchcouchcrashcrushdeathearthfetchfifthflashfleshflushforthfreshgnashhatchlatchlunchlurchmarshmatchmorphneathninthnorthpeachperchplushpoochporchreachsixthslushsoothsouthteachtenthtoothtruthyouth
A vegetable that starts with h would be horseradish. That is the only vegetable that I can find that begins with the letter h.
7-letter words ending with H:anguishbeneathbeseechblemishboorishbrutishcherisheyelashfurnishgarnishimpeachmannishmenorahnourishpiggishprudishreddishrematchscratchselfishstealthsunfishtarnishunearthunleashunlatchvarnishwarpath
One is through.
Warpath, seventh, anguish.
SELFISH.
One word that fits this definition is broth.
Carvagh
Five letter words starting with H and ending in H are:hatchhaughfeathheighheuchheughhitchhoochhunchhutchhumphhorahhotchharsh, heath.
Some five letter words ending in H are:batchbeachbelchbirchbirthblushbrashbrushbunchcatchcoachconchcouchcrashcrushdeathearthfetchfifthflashfleshflushforthfreshgnashhatchlatchlunchlurchmarshmatchmorphneathninthnorthpeachperchplushpoochporchreachsixthslushsoothsouthteachtenthtoothtruthyouth
Pharaoh, 7 letters only.
Hominy?
spinach
Horseraddish