Term for a set that is not immune
If a set $A$ is Turing reducible to a set $B$ then we say that $B$ computes $A$.

Every noncomputable set $A$ computes an immune set, namely $\hat A = \{\sigma: \sigma \text{ is a prefix of }A\}$.

(If $A$ is a set of strings then we first turn it onto a [i.e. replace it by a Turing equivalent] set of integers or equivalently an infinite binary sequence before applying the hat operation.)

For instance if $A$ is the set of prime numbers then we may write
$$A=A(0)A(1)A(2)\dots=00110101000101000101\dots$$
and then $\sigma=0011010$ is a prefix of $A$ whereas $010011$ is not.

Therefore the complexity class of sets that compute only non-immune sets is just the computable sets $\mathsf{R}$.