An $m=1$ lopsided asymmetry is common in disc galaxies. Here, we investigate the excitation of an $m=1$ lopsidedness in host galaxies during minor mergers (mass ratio 1:10) while choosing a set of minor merger models (with varying orbital configurations, morphology of the host galaxy) from the GalMer library of galaxy merger simulations. We show that a minor merger triggers a prominent $m=1$ lopsidedness in the stars of the host galaxy. The strength of the $m=1$ lopsidedness undergoes a transient amplification phase after each pericentre passage of the satellite, in concordance with past findings of excitation of an $m=1$ lopsidedness due to tidal encounters. However, once the merger happens, and the post-merger remnant readjusts itself, the lopsidedness fades away in short time-scale ($sim 500-850$ Myr). Furthermore, a delayed merger can drive a prolonged ($sim 2$ Gyr) lopsidedness in the host galaxy. We demonstrate that the $m=1$ lopsidedness rotates with a well-defined pattern speed. The measured pattern speed is much slower than the $m=2$ bar pattern speed, and is retrograde with respect to the bar. This gives rise to a dynamical scenario where the Inner Linblad resonance (ILR) of the $m=1$ lopsidedness falls in between the corotation (CR) and the Outer Linblad resonance (OLR) of the $m=2$ bar mode. A kinematic lopsidedness also arises in the host galaxy, and the resulting temporal variation closely follows that of the density lopsidedness. The minor merger also triggers a transient off-centred stellar disc-dark matter halo configuration due to the tidal encounter with the satellite.
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