Corrections to electroweak precision observables from mixings of an exotic vector boson in light of the CDF $W$-mass anomaly

We enumerate various effective couplings that contribute to the mixings between an exotic vector boson $Z^{\mathrm{\prime }}$ and the neutral electroweak vector bosons. The miscellaneous mixing patterns can be evaluated perturbatively. The effective oblique parameters $S^{\mathrm{\prime }}$, $T^{\mathrm{\prime }}$, and $U^{\mathrm{\prime }}$ are calculated to compare with the electroweak precision test results. With the contributions to the non-negligible $U^{\mathrm{\prime }}$ parameter from the ${ϵ}_{B,W}$ parameters and the aid of some other parameters to cancel the negative $T^{\mathrm{\prime }}$, the recent CDF $W$-mass anomaly can therefore be explained.