hospital
hot
Halt.
heartless
hate
Haste is a synonym for in a hurry. It begins with the letter h.
Ill-fortune, hardship, and pileup.
Something to do with help: helping, helpfulness
Heavenly is a synonym for tasty.
Another word (synonym) for the word "unpleasant" that starts with an "h" is "horrific". Or at least that is what I could think of.
Active is a synonym for athletic. It starts with the letter a.
Substantial is a synonym for big that starts with S.
Boost is a synonym for the word lift and starts with the letter B.