Definition
The skew polynomial algebra over $\mathbb{C}$ is defined as $\mathbb{C}\langle x,y\rangle/(xy-yx+x)$ or alternatively as $\mathbb{C}[x,y,\sigma]$, where $\sigma$ is the automorphism on $\mathbb{C}[x]$ sending $x$ to $x+1$.
Question
Does anyone know a reference where the Hochschild cohomology of $\mathbb{C}[x,y,\sigma]$ is calculated?
It is not very difficult to compute it by hand! I do not recall seeing a reference for this — in the literature dimension three seems to be the lower limit of difficulty that's deemed publisheable :-)
Notice that your algebra is the enveloping algebra of the nonabelian Lie algebra of dimension $2$. This observation gives you a simple resolution with which to do the calculation: the Chevalley-Eilenberg complex. On the other hand, your algebra has a grading ($x$ in degree $1$, $y$ in degree $0$) and the complex you have to work with is homogeneous with respect to that grading, so this cuts down the work a lot!
An extra tip is that grading is in fact an inner one: the derivation $a\in A\mapsto [y,a]\in A$ of the algebra is diagonalizable, its eigenvalues are integers, and its eigenspaces are precisely the homogeneous components of the algebra. One can use this to shopw that in fact the only relevant part of the complex is the one in grading zero, and then the calculation is almost trivial.
If you really cannot do this, I can write down the details.