A popular class of extensions of the Standard Model (SM) are models of a new Abelian gauge boson $X$, called dark or hidden photon, that kinetically mixes with the SM photon.
We revisit the matching procedure of kinetic mixing terms in the electroweak symmetric phase to the ones in the broken phase. Our central finding is that in order to obtain the correct matching prescription one has to take into account mixing of the hidden photon with the neutral component of the weak $SU(2)_L$ bosons. This mixing is generated by a dimension-six operator and, in theories where $SU(2)_L$ multiplets are charged under the novel Abelian gauge group, is necessarily induced at the one-loop level. We illustrate this matching procedure for the loop-generated kinetic mixing in $U(1)_{L_\mu-L_\tau}$. Furthermore, we show how to obtain general expressions for the Higgs decay amplitudes to two neutral vector bosons from the vacuum polarisation amplitudes via the low-energy theorems. As an application, we derive general expression for the branching ratios of the decays $h\to\gamma X$ and $h\to XX$ in $U(1)_{B-L}$.