In this paper the author introduces a new cohomology theory called the secondary cohomology. More precisely, if \(A\) and \(B\) are \(k\)-algebras, \(B\) is commutative, \(\varepsilon\colon B\to A\) is a morphism of \(k\)-algebras such that \(\varepsilon(B)\subset\mathcal Z(A)\), and \(M\) is an \(A\)-bimodule, he constructs cohomology groups \(H^n((A,B,\varepsilon);M)\). For more properties of this secondary Hochschild cohomology see M. D. Staic and A. Stancu [Homology Homotopy Appl. 17, No. 1, 129-146 (2015; Zbl 1346.16004)]. Using this Hochschild-like cohomology the author describes simultaneous deformation of the product and of the \(B\)-algebra structure on \(A[[t]]\). Also the author shows that these deformations have the property that the natural projection \(A[[t]]\to A\) is a morphism of \(B\)-algebras.