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We study the hybrid classical-quantum version of the channel coding problem for the famous Gel’fand–Pinsker channel. In the classical setting for this channel the conditional distribution of the channel output given the channel input is a function of a random parameter called the channel state. We study this problem when a rate limited version of the channel state is available at the encoder for the classical-quantum Gel’fand–Pinsker channel. We establish the capacity region for this problem in the information-spectrum setting. The capacity region is quantified in terms of spectral-sup classical mutual information rate and spectral-inf quantum mutual information rate. [ABSTRACT FROM AUTHOR]