What exactly is 'mesquite' flavour?

The wood of the mesquite tree is used as a 'smoking wood' (barbequing) usually for beef and gives a smokey, sometimes bitter flavour to enhance the food. Hickory give a similar flavour to the meat, but not quite as strong.

So, the answer would be wood-smoke flavour.

Mesquite

Mesquite is the common name for the genus Prosopis spp and there are over 40 species of small leguminous trees. They are native to the southwestern United States and Mexico (except for creeping mesquite, which is invasive in southern California). The mesquite originates from the Tamaulipan mezquital ecoregion, in the deserts and xeric shrublands biome, located in the southern United States and northeastern Mexico. It covers an area of 141,500 km2 (54,600 sq mi), encompassing a portion of the Gulf Coastal Plain in southern Texas, northern Tamaulipas, northeastern Coahuila, and part of Nuevo León. As a legume, it is one of the few sources of fixed nitrogen in the desert habitat.

https://en.wikipedia.org/wiki/Mesquite