Heegaard genus, degree-one maps, and amalgamation of 3-manifolds

Let $M=W{\cup }_{T}V$ be an amalgamation of two compact 3-manifolds along a torus, where $W$ is the exterior of a knot in a homology sphere. Let $N$ be the manifold obtained by replacing $W$ with a solid torus such that the boundary of a Seifert surface in $W$ is a meridian of the solid torus. This means that there is a degree-one map $f:M\to N$, pinching $W$ into a solid torus while fixing $V$. We prove that $g\left(M\right)⩾g\left(N\right)$, where $g\left(M\right)$ denotes the Heegaard genus. An immediate corollary is that the tunnel number of a satellite knot is at least as large as the tunnel number of its pattern knot.