I have often seen the statement that HyperKahler (HK) manifolds have torsionfree connections. In general relativity, however, one is usually taught that the connection is something that you can "choose". You might choose to use the LeviCivita connection or not. Is it not possible for a HK manifold to choose a connection with torsion?

You may be understandably confused by a terminological inconsistency. People study socalled "HyperKähler manifolds with torsion" (a.k.a. HKT manifolds), which by definition have
such that: the tensor $g(\cdot,T(\cdot,\cdot))$ defined by the torsion $T$ of $\nabla$ is totally antisymmetric (a 3form). The irony, observed in the very review of their introduction, is that unless $T=0$, these manifolds are not HyperKähler in the classical sense of having their (LeviCivita) holonomy contained in $\operatorname{Sp}(n)$. 

