  We argue that the lensing power spectrum of astrometric shift (lensing shift
power spectrum) is a powerful tool of the clustering property of dark matter on
subgalactic scales. First we give the formalism to probe the nature of dark
matter by using the lensing shift power spectrum. Then, leveraging recent
measurements of the lensing shift power spectrum on an angular scale of
approximately $1~$arcsec towards the gravitationally lensed quasar
MG$\,$J0414+0534 at the redshift of $z_S=2.639$, we place constraints on the
mass of warm dark matter (WDM) particles $m_{\rm WDM}$ and their fraction in a
mixed dark matter (MDM) model $r_{\rm WDM}$, in which WDM and cold dark matter
coexist. Although the constraint derived from the above single lensing system
is not as strong as the existing constraints, as we show in this paper, the
lensing shift power spectrum has a great potential to obtain much tighter
constraints on WDM and MDM models through future observations, highlighting the
importance of well-controlled systematic error considerations for achieving
enhanced precision.
