  We obtain that Sarnak's M\"{o}bius Disjointness Conjecture holds for product
flows between affine linear flows on compact abelian groups of zero topological
entropy and a class of rigid dynamical systems. To prove this, we show an
estimate for the average value of the product of the M\"obius function and any
polynomial phase over short intervals and arithmetic progressions
simultaneously. In addition, we prove that the logarithmically averaged
M\"obius Disjointness Conjecture holds for the product flow between any affine
linear flow on a compact abelian group of zero entropy and any rigid dynamical
system. As an application, we show that the logarithmically averaged M\"obius
Disjointness Conjecture holds for every Lipschitz continuous skew product
dynamical system on $\mathbb{T}^2$ over a rotation of the circle.
