The List
There are 253 4 letter words with H as the first letter. The highest scoring is HIZZ at 25 points in Scrabble. 3 of them contain no vowel at all: HWYL, HYMN, HYPS.
HAAFHAARHABUHACKHADEHADJHADSHAEDHAEMHAENHAESHAETHAFFHAFTHAGGHAGSHAHAHAHSHAIKHAILHAINHAIRHAJIHAJJHAKAHAKEHAKUHALEHALFHALLHALMHALOHALTHAMEHAMSHANDHANGHANKHANTHAOSHAPSHAPUHARDHAREHARKHARLHARMHARNHAROHARPHARTHASHHASKHASPHASSHASTHATEHATHHATSHAUDHAUFHAULHAUTHAVEHAWKHAWMHAWSHAYSHAZEHAZYHEADHEALHEAPHEARHEATHEBEHECHHECKHEEDHEELHEFTHEHSHEIDHEILHEIRHELDHELEHELMHELOHELPHEMEHEMPHEMSHENDHENSHENTHEPSHEPTHERBHERDHEREHERLHERMHERNHEROHERSHERYHESPHESTHETEHETHHETSHEWNHEWSHEYSHICKHIDEHIEDHIESHIGHHIKEHILAHILDHILIHILLHILTHIMSHINDHINGHINSHINTHIOIHIPSHIPTHIREHISHHISNHISSHISTHITSHIVEHIYAHIZZHOASHOAXHOBOHOBSHOCKHODSHOEDHOESHOGGHOGHHOGSHOHAHOHSHOIKHOKAHOKEHOKIHOLDHOLEHOLKHOLMHOLPHOLSHOLTHOLYHOMAHOMEHOMSHOMYHONDHONEHONGHONKHONSHOODHOOFHOOKHOONHOOPHOORHOOTHOPEHOPSHORAHORIHORNHORSHOSEHOSSHOSTHOTEHOTSHOUFHOURHOUTHOVEHOWEHOWFHOWKHOWLHOWSHOYAHOYSHUBSHUCKHUEDHUERHUESHUFFHUGEHUGSHUGYHUHUHUIAHUICHUISHULAHULEHULKHULLHUMAHUMFHUMPHUMSHUNGHUNHHUNKHUNSHUNTHUPSHURLHURTHUSHHUSKHUSOHUSSHUTSHWANHWYLHYEDHYENHYESHYKEHYLAHYLEHYMNHYPEHYPOHYPSHYTE