I can list some words with H in the beginning:
- ha
- haberdasher
- had
- hail
- hair
- hale
- hallowed
- hammock
- handy
- haphazard
- happen
- happy
- hard
- hare
- hark
- harm
- harness
- harrowing
- harry
- haste
- hat
- hate
- hazard
- hazelnut
- hazard
- haze
- hazy
- head
- heal
- health
- heap
- hear
- heart
- heat
- heave
- heaven
- heavy
- heed
- heel
- heinous
- helium
- hell
- help
- hem
- her
- herald
- herb
- herbivore
- hibernate
- hiccup
- hideous
- hieroglyphics
- hill
- hillside
- hilly
- him
- hint
- hip
- hippo
- hire
- his
- history
- ho
- hoe
- hoard
- hold
- hole
- hollow
- home
- homogenous
- hone
- hook
- hoop
- hoot
- hope
- horde
- horn
- hose
- horrid
- hotel
- house
- hovel
- hover
- hug
- huge
- human
- humiliate
- humorous
- hump
- hunger
- hungry
- hunk
- hunt
- hurl
- hurt
- hydrangea
- hydroelectric
- hydrology
- hydrogen
- hysterectomy
- hysterical