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We prove smoothing properties of nonlocal transition semigroups associated toa class of stochastic differential equations (SDE) driven by additive pure-jumpL\'evy noise. In particular, we assume that the L\'evy process driving the SDEis the sum of a subordinated Wiener process and of an independent arbitraryL\'evy process, that the drift coefficient is continuous (but not necessarilyLipschitz continuous) and grows not faster than a polynomial, and that the SDEadmits a Feller weak solution. By a combination of probabilistic and analyticmethods, we provide sufficient conditions for the Markovian semigroupassociated to the SDE to be strong Feller and to map $L_p$ to continuousbounded functions. A key intermediate step is the study of regularizingproperties of the transition semigroup associated to the subordinated Wienerprocess in terms of negative moments of the subordinator.

On smoothing properties of transition semigroups associated to a class of SDEs with jumps