  The relativistic recoil contributions to the Uehling corrections are
revisited. We consider a controversy in recent calculations based on different
approaches including Breit-type and Grotch-type calculations. We have found
that calculations of those authors were in fact done in different gauges and in
some of those gauges contributions the retardation and two-photon-exchange
effects were missed. We have evaluated such effects and obtained a consistent
result from those approaches. We present a correct expression for the
Grotch-type approach which produces a correct gauge-invariant result. We also
consider a finite-nuclear-size correction for the Uehling term. The results are
presented for muonic hydrogen and deuterium atoms and for muonic helium-3 and
helium-4 ions.
