A steeper fall of fusion excitation function, compared to the predictions of coupled-channels models, at energies below the lowest barrier between the reaction partners, is termed as deep sub-barrier fusion hindrance. This phenomenon has been observed in many symmetric and nearly-symmetric systems. Different physical origins of the hindrance have been proposed. This work aims to study the probable effects of direct reactions on deep sub-barrier fusion cross sections. Fusion (evaporation residue) cross sections have been measured for the system $^{19}$F+$^{181}$Ta, from above the barrier down to the energies where fusion hindrance is expected to come into play. Coupled-channels calculation with standard Woods-Saxon potential gives a fair description of the fusion excitation function down to energies $simeq 14%$ below the barrier for the present system. This is in contrast with the observation of increasing fusion hindrance in asymmetric reactions induced by increasingly heavier projectiles, textit{viz.} $^{6,7}$Li, $^{11}$B, $^{12}$C and $^{16}$O. The asymmetric reactions, which have not shown any signature of fusion hindrance within the measured energy range, are found to be induced by projectiles with lower $alpha$ break-up threshold, compared to the reactions which have shown signatures of fusion hindrance. In addition, most of the $Q$-values for light particles pick-up channels are negative for the reactions which have exhibited strong signatures of fusion hindrance, textit{viz.} $^{12}$C+$^{198}$Pt and $^{16}$O+$^{204,208}$Pb. Thus, break-up of projectile and particle transfer channels with positive $Q$-values seem to compensate for the hindrance in fusion deep below the barrier. Inclusion of break-up and transfer channels within the framework of coupled-channels calculation would be of interest.
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