Yes, MESQUITE is a Scrabble word!

Letter values in Scrabble US: M 3, E 1, S 1, Q 10, U 1, I 1, T 1, E 1.

Is it a valid word?

MESQUITE is a valid word in the official Scrabble dictionary (US), the UK Scrabble Dictionary and Words with Friends (ENABLE).

any of several small spiny trees or shrubs of the genus Prosopis having small flowers in axillary cylindrical spikes followed by large pods rich in sugar