The coaction and competition between the ferroelectric field effect and the strain effect in Pr0.5Ca0.5MnO3 (PCMO) film/0.67Pb(Mg1/3Nb2/3)O3-0.33PbTiO3 crystal heterostructures were studied. Based on different types of resistance-electric field hysteresis loops at various temperatures, it is clearly identified that the strain effect dominates over the ferroelectric field effect for temperature T above the charge-ordering temperature TCO of PCMO. With the strong localization of charge carriers for T < TCO, the ferroelectric field effect strongly competes with the strain effect and finally dominates over the latter for T < 0.8TCO. Moreover, the poling-induced strain effect is considerably enhanced by a magnetic field, demonstrating the important role of the phase separation in understanding the strain effect in such heterostructures.