Minor league hitting coordinator Lamar Johnson will take over for Hudgens.
The 57-year-old Hudgens was in his fourth season as the Mets' hitting coach. The Mets began the day with a .237 batting average, ranking among the worst in the majors.
New York has lost nine of its last 12 games and scored a total of only 32 runs in that span.
The Mets have tried to implement a plan of patience at the plate in the majors and minors, but the results have not shown.
"When you struggle the way we've struggled offensively, a finger, obviously, gets pointed. A lot of times that finger gets pointed unfairly," Mets star David Wright said.
Valverde gave up four runs while getting only two outs in the eighth and ninth innings as the Pirates took the lead. The 36-year-old righty was 1-1 with a 5.66 earned-run average.
The Tampa Bay Rays put outfielder Brandon Guyer on the 15-day disabled list because of a broken left thumb and recalled right-hander Alex Colome from single-A Charlotte (Fla.).