We have investigated the contribution of the Higher Twist (HT) Feynman diagrams to the large-p(T) inclusive rho(L) meson production cross-section in proton-proton and proton-antiproton collisions and we discuss the phenomenological consequences of possible HT contributions to cross-sections. To extract HT subprocesses from Leading Twist (LT) background, we use various rho(L) meson distribution amplitudes (DAs). In the numerical calculations, the dependencies of the HT contribution on the transverse momentum and the rapidity are discussed with special emphasis on DAs. Analysis of our results shows that HT contributions decrease more rapidly than LT contributions with increasing p(T). The preceding results demonstrate that HT contributions must be considered especially in the region of low p(T) and the HT contribution to the cross-section depends on the choice of the meson DAs.