The structure $L_{\omega_1^{CK}}$ consists of only HYP sets (I believe) and HYP in this structure is the same as the actual hyperaritmetic sets. Now if I move to the structure $L_{\omega_1^{CK}}[a]$ where $a$ is some non-hyperarithmetic real I add my guess is that HYP remains unaltered simply because HYP has a bottom up definition and won't be affected by the addition of new sets as long as one doesn't move to a non-$\omega$ model or the like.

Is this correct? Any more formal demonstration?