Mendy was given the chance to atone for his mistake and pull a goal back in the 15th minute when City won a penalty for a foul on Nick Barmby, but Crewe keeper Steve Collis saved the Frenchman's strike.
Mendy then gave away two penalties at the other end before half time. Nicky Maynard and Calvin Zola, for his hat-trick, successfully put them away.
Phil Brown brought on 11 substitutes at half-time, but they couldn't overturn the 4-0 scoreline - City's first defeat of this pre-season.
City line-up (first half): Boaz Myhill; Bernard Mendy, Liam Cooper, Wayne Brown, Nathan Doyle; Richard Garcia, George Boateng, Ian Ashbee, Péter Halmosi; Caleb Folan, Nick Barmby.
City line-up (second half): Matt Duke; Ryan France, Matt Plummer, Michael Turner, Sam Ricketts; Craig Fagan, Dean Marney, Nicky Featherstone, Bryan Hughes; Geovanni, Dean Windass.