Maybe, at least in part
In "Heroes' Day: Catalyst", Hawkmoth seems to have a plan by which he akumatizes Lila Rossi into Volpina, and has her create an illusion to demoralize the people of Paris.
He then akumatizes Nathalie Sancoeur into Catalyst, with her cooperation. She gives him some sort of boost, allowing him to create multiple akuma, which seem to target his previous victims and re-akumatize them into versions of their old forms.
On the one hand, the fact that all the Catalyst-akumatized villains took on the exact same forms that they had previously suggests that Hawkmoth did not control what form they took. Otherwise he would have chosen powers that the heroes had not dealt with before, or at least they wouldn't have had any default form.
Similarly, the fact that he specifically had to get Lila angry enough to turn her into Volpina to create the illusion strongly suggests that he couldn't just choose to give her power to Nathalie, since that would have been the easier and more obvious choice.
On the other hand, the fact that his whole plan hinged on turning Nathalie into Catalyst suggests that he was able to choose to give her the Catalyst powers.
One possible explanation is that Hawkmoth did not control Natalie's power. Rather, he already knew that Nathalie would have that power set, because he'd akumatized her before, possibly as practice. It's also possible that there's a finite list of powers, and that Hawkmoth and Nathalie knew what power she'd get based on her character.
Further evidence is provided by Season 3, Episode 1, in which he akumatizes Lila. He says "I have some new powers for you, Chameleon" again implying that he can choose which powers to give. In addition, she doesn't get akumatized into Volpina this time. On the other hand, she still gets somewhat illusion-based powers (although she seemingly can also copy powers), in a similar vein to Volpina, and Hawkmoth doesn't choose to give her a stronger power set such as that of Catalyst or Style Queen, perhaps again suggesting that he doesn't have total control over which powers he grants.