Is a degree-$d$ pseudo distribution always a relaxation?
To see that this is a relaxation, all you have to check is that each feasible solution to the first problem gives rise to a feasible solution to the second problem. All you have to do is, given $x \in K$, take an actual distribution $D$ concentrated on $x$. (This only works if in the definition of $K$ you replace $\mathbb{R}^n$ with $\{0,1\}^n$; your formulation doesn't make too much sense.)

When $d = n$, the conditions guarantee that $D$ is an actual distribution. If the objective function $p$ is concave then the optimal distributions are singletons, and so both programs have the same value. Otherwise it could be that for some distribution $D$ supported on $K$ it holds that $\mathbb{E}_{X\sim D}[p(X)] &lt; \min_x p(x)$.