In this work, we study a class of skew cyclic codes over the ring $R:=\mathbb{Z}_4+v\mathbb{Z}_4,$ where $v^2=v,$ with an automorphism $θ$ and a derivation $Δ_θ,$ namely codes as modules over a skew polynomial ring $R[x;θ,Δ_θ],$ whose multiplication is defined using an automorphism $θ$ and a derivation $Δ_θ.$ We investigate the structures of a skew polynomial ring $R[x;θ,Δ_θ].$ We define $Δ_θ$-cyclic codes as a generalization of the notion of cyclic codes. The properties of $Δ_θ$-cyclic codes as well as dual $Δ_θ$-cyclic codes are derived. As an application, some new linear codes over $\mathbb{Z}_4$ with good parameters are obtained by Plotkin sum construction, also via a Gray map as well as residue and torsion codes of these codes.