The effect of a periodic train of short gas-puff pulses on the rotation frequency and amplitude of drift-tearing modes has been studied in ADITYA/ADITYA-U tokamak. The short gas puffs, injecting approximately similar to 10(17)-10(18) molecules of fuel gas (hydrogen) at one toroidal location, are found to concomitantly decrease the drift-tearing mode rotation frequency and the mode amplitude during the period of injection and then recover back to its initial values when the gas pulse is over. This leads to a periodic modulation of the rotation frequency and amplitude of the drift-tearing modes that is correlated with the periodicity of the gas pulse injection. The underlying mechanism for this change in the mode characteristic appears to be related to gas puff induced change in the radial profile of the plasma pressure in the edge region that brings about a reduction in the diamagnetic drift frequency. Detailed experimental measurements and BOUT++ code simulations support such a reduction in diamagnetic drift frequency. Our results reveal a close interaction between the edge dynamics and core MHD phenomena in a tokamak that could help us better understand the rotation dynamics and amplitude pulsations of magnetic islands.