The List
There are 479 5 letter words with H as the first letter. The highest scoring is HUZZY at 29 points in Scrabble. 3 of them contain no vowel at all: HWYLS, HYMNS, HYPHY.
HAAFSHAARSHABITHABLEHABUSHACEKHACKSHADALHADEDHADESHADJIHADSTHAEMSHAETSHAFFSHAFISHAFIZHAFTSHAGGSHAHASHAICKHAIKAHAIKSHAIKUHAILSHAILYHAINSHAINTHAIRSHAIRYHAITHHAJESHAJISHAJJIHAKAMHAKASHAKEAHAKESHAKIMHAKUSHALALHALEDHALERHALESHALFAHALFSHALIDHALLOHALLSHALMAHALMSHALONHALOSHALSEHALTSHALVAHALVEHAMALHAMBAHAMEDHAMESHAMMYHAMZAHANAPHANCEHANCHHANDSHANDYHANGIHANGSHANKSHANKYHANSAHANSEHANTSHAOLEHAOMAHAPAXHAPLYHAPPYHAPUSHARAMHARDSHARDYHAREDHAREMHARESHARIMHARKSHARLSHARMSHARNSHAROSHARPSHARPYHARRYHARSHHARTSHASHYHASKSHASPSHASTAHASTEHASTYHATCHHATEDHATERHATESHATHAHAUDSHAUFSHAUGHHAULDHAULMHAULSHAULTHAUNTHAUSEHAUTEHAVENHAVERHAVESHAVOCHAWEDHAWKSHAWMSHAWSEHAYEDHAYERHAYEYHAYLEHAZANHAZEDHAZELHAZERHAZESHEADSHEADYHEALDHEALSHEAMEHEAPSHEAPYHEARDHEAREHEARSHEARTHEASTHEATHHEATSHEAVEHEAVYHEBENHEBESHECHTHECKSHEDERHEDGEHEDGYHEEDSHEEDYHEELSHEEZEHEFTEHEFTSHEFTYHEIDSHEIGHHEILSHEIRSHEISTHEJABHEJRAHELEDHELESHELIOHELIXHELLOHELLSHELMSHELOSHELOTHELPSHELVEHEMALHEMESHEMICHEMINHEMPSHEMPYHENCEHENDSHENGEHENNAHENNYHENRYHENTSHEPARHERBSHERBYHERDSHERESHERLSHERMAHERMSHERNSHERONHEROSHERRYHERSEHERTZHERYEHESPSHESTSHETESHETHSHEUCHHEUGHHEVEAHEWEDHEWERHEWGHHEXADHEXEDHEXERHEXESHEXYLHEYEDHIANTHICKSHIDEDHIDERHIDESHIEMSHIGHSHIGHTHIJABHIJRAHIKEDHIKERHIKESHIKOIHILARHILCHHILLOHILLSHILLYHILTSHILUMHILUSHIMBOHINAUHINDSHINGEHINGSHINKYHINNYHINTSHIOISHIPLYHIPPOHIPPYHIREDHIREEHIRERHIRESHISSYHISTSHITCHHITHEHIVEDHIVERHIVESHIZENHOAEDHOAGYHOARDHOARSHOARYHOASTHOBBYHOBOSHOCKSHOCUSHODADHODJAHOERSHOGANHOGENHOGGSHOGHSHOHEDHOICKHOIKSHOINGHOISEHOISTHOKASHOKEDHOKESHOKEYHOKISHOKKUHOKUMHOLDSHOLEDHOLESHOLEYHOLKSHOLLAHOLLOHOLLYHOLMSHOLONHOLTSHOMASHOMEDHOMERHOMESHOMEYHOMIEHOMMEHOMOSHONANHONDAHONDSHONEDHONERHONESHONEYHONGIHONGSHONKSHONKYHONORHOOCHHOODSHOODYHOOEYHOOFSHOOKAHOOKSHOOKYHOOLYHOONSHOOPSHOORDHOORSHOOSHHOOTSHOOTYHOOVEHOPEDHOPERHOPESHOPPYHORAHHORALHORASHORDEHORISHORMEHORNSHORSEHORSTHORSYHOSEDHOSELHOSENHOSERHOSESHOSEYHOSTAHOSTSHOTCHHOTELHOTENHOTLYHOTTYHOUFFHOUFSHOUGHHOUNDHOURIHOURSHOUSEHOUTSHOVEAHOVEDHOVELHOVENHOVERHOVESHOWBEHOWDYHOWESHOWFFHOWFSHOWKSHOWLSHOWREHOWSOHOXEDHOXESHOYASHOYEDHOYLEHUBBYHUCKSHUDNAHUDUDHUERSHUFFSHUFFYHUGERHUGGYHUHUSHUIASHULASHULESHULKSHULKYHULLOHULLSHULLYHUMANHUMASHUMFSHUMICHUMIDHUMORHUMPHHUMPSHUMPYHUMUSHUNCHHUNKSHUNKYHUNTSHURDSHURLSHURLYHURRAHURRYHURSTHURTSHUSHYHUSKSHUSKYHUSOSHUSSYHUTCHHUTIAHUZZAHUZZYHWYLSHYDRAHYDROHYENAHYENSHYINGHYKESHYLASHYLEGHYLESHYLICHYMENHYMNSHYNDEHYOIDHYPEDHYPERHYPESHYPHAHYPHYHYPOSHYRAXHYSONHYTHE