The List
There are 155 4 letter words with H as the last letter. The highest scoring is FIQH at 19 points in Scrabble, though Words With Friends prefers QOPH at 19. 4 of them contain no vowel at all: LYCH, MYTH, SYPH, WYCH.
AMAHANKHARCHARGHAYAHBACHBAGHBASHBATHBETHBISHBLAHBOOHBOSHBOTHBUSHCAPHCASHCOCHCOSHCOTHCUSHDASHDICHDISHDOSHDOTHDUSHEACHEATHEECHEISHETCHEUGHFASHFIQHFISHGASHGATHGOSHGOTHGUSHHASHHATHHECHHETHHIGHHISHHOGHHUNHHUSHINCHITCHJOSHKAPHKESHKISHKITHKOPHKUEHLAKHLASHLATHLECHLICHLITHLOCHLOSHLOTHLUSHLYCHMACHMASHMATHMESHMETHMICHMOCHMOSHMOTHMUCHMUSHMWAHMYTHNACHNESHNIGHNISHNOAHNOSHOATHODAHOKEHOPAHOUCHOUPHPASHPATHPECHPEGHPHOHPISHPITHPOOHPOSHPRUHPUGHPUSHQOPHRACHRASHRATHRESHRICHROCHRUKHRUSHRUTHSASHSECHSESHSHAHSICHSIDHSIGHSINHSITHSOPHSOTHSUCHSUGHSUKHSYPHTACHTANHTASHTATHTECHTETHTICHTOPHTOSHTUSHUMPHVUGHWASHWICHWISHWITHWYCHYEAHYECHYODHYOGHYUCH