  Manipulation of the valley degree of freedom provides a novel paradigm in
quantum information technology. Here, through first-principles calculations and
model analysis, we demonstrate that monolayer Cr$_2$COOH MXene is a promising
candidate material for valleytronics applications. We reveal that Cr$_2$COOH is
a ferromagnetic semiconductor and harbors valley features. Due to the
simultaneous breaking inversion symmetry and time-reversal symmetry, the
valleys are polarized spontaneously. Moreover, the valley polarization is
sizeable in both the valence and conduction bands, benefiting the observation
of the anomalous valley Hall effect. More remarkably, the valley splitting can
be effectively tuned by the magnetization direction, strain and ferroelectric
substrate. More interestingly, the ferroelectric substrate Sc$_2$CO$_2$ can not
only regulate the MAE, but also tune valley polarization state. Our findings
offer a practical way for realizing highly tunable valleys by multiferroic
couplings.
