I don't recall ever seeing Long Island fresh sweet "mush melons" for sale. The much more common name for the species, which includes many cultivars, is "muskmelon," owing to the aroma of the mature fruit; "mush," by many accounts, is an old corruption. This particular cultivar, considered in cross-section, you'd probably call a cantaloupe.
Elliniki Agora Fruit & Vegetables
3212 30th Ave. (31st-32nd Sts.), Astoria, Queens
718-728-0751