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Dynamical chiral symmetry breaking (D$\chi$SB) is studied within($2+1$)-dimensional QED with $N$ four-component fermions. The leading andnext-to-leading orders of the $N$ expansion were computed exactly inRefs.~[\onlinecite{Gusynin:2016som,Kotikov:2016prf}] in an arbitrary non-localgauge. In this addendum to [\onlinecite{Kotikov:2016prf}], we show that theresummation of the wave-function renormalization constant at the level of thegap equation yields a {\it complete} cancellation of the gauge dependence ofthe critical fermion flavour number resulting in: $N_c= 2.8469$, which is suchthat D$\chi$SB takes place for $N<N_c$. The result is in full agreement withone of Ref.~[\onlinecite{Gusynin:2016som}].

Addendum to "Critical behaviour of ($2+1$)-dimensional QED: $1/N_f$-corrections in an arbitrary non-local gauge"