Quite unlikely. The many solutions to Deligne's conjecture for Hochschild cohomology show that the Hochschild complex is an $E_2$-algebra, and this is the best one can get. It is known that the cohomology of an $E_\infty$-algebra carries an action of the Steenrod algebra. If you wish, this is why the standard group cohomology is an algebra over the Steenrod algebra, since it is the cohomology of the (singular) cochains on a space, which is an $E_\infty$-algebra. The homology of an $E_2$-algebra has an $Sq^2$ in characteristic $2$ and no other Steenrod operation otherwise.

Quite unlikely. The many solutions to Deligne's conjecture for Hochschild cohomology show that the Hochschild complex is an $E_2$-algebra, and this is the best one can get. It is known that the cohomology of an $E_\infty$-algebra carries an action of the Steenrod algebra. If you wish, this is why the standard group cohomology is an algebra over the Steenrod algebra, since it is the cohomology of the (singular) cochains on a space, which is an $E_\infty$-algebra. The homology of an $E_2$-algebra has a very low-dimensional $p$-power operation, where $p$ is the cgaracteristic of the ground field, and that's essentially all, see the references below.

PD I've edited this comment to correct some inaccuracies.