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By using the gauge-invariant but path-dependent, variables formalism, weconsider a recently proposed topologically massive $U{\left( 1 \right)_{\calW}} \times U{(1)_{\cal Y}}$ Chern-Simons-Higgs theory in $2+1$ dimensions. Inparticular, we inspect the impact of a Chern-Simons mixing term between twoAbelian gauge fields on physical observables. We pursue our investigation byanalysing the model in two different situations. In the first case, where weintegrate out the massive excitation and consider an effective model for themassless field, we show that the interaction energy contains a linear termleading to the confinement of static charge probes along with a screeningcontribution. The second situation, where the massless field can be exactlyintegrated over with its constraint duly taken into account, the interestingfeature is that the resulting effective model describes a purely screeningphase, without any trace of a confining regime.

Aspects of screening and confinement in a topologically massive $U{\left( 1 \right)_{\cal W}} \times U{(1)_{\cal Y}}$ Chern-Simons-Higgs theory