The phrase "to lift" means to pick something up. There are several different words that can be substituted for it. Synonyms for it that begins with the letter "H" would be heave, heft, hike, and hoist.
Boost is a synonym for the word lift and starts with the letter B.
rise
hot
hospital
Halt.
heartless
hate
Haste is a synonym for in a hurry. It begins with the letter h.
Lift is a synonym for elevate
Ill-fortune, hardship, and pileup.
Something to do with help: helping, helpfulness
Heavenly is a synonym for tasty.