is still, to some extent, in its infancy. Pei, Xu and Yin [@peistochastic2017] established an averaging principle for a system of SPDEs that have a slow component driven by an additive fractional noise and a fast component driven by fast-varying diffusion. Pei et al. [@pei2018sd; @Averaging2018pei] examined averaging principles for SPDEs driven by an additive fractional noise with two-time-scale Markovian switching processes. But, till now, in multiplicative fractional noise case, only the averaging results for SDE cases were obtained. Hairer and Li [@Hairer2019averaging] considered slow-fast systems where the slow system is driven by fBm and proved the convergence to the averaged solution took place in probability. Very recently, Pei, Inahama and Xu [@pei2020averaging] answered affirmatively that an averaging principle still holds for fast-slow mixed SDEs if disturbances involve both Bm and long-range dependence modeled by fBm $H\in(\frac{1}{2},1)$ in the mean square sense.

To the best of our knowledge, the second part of our paper is the first attempt to study stochastic averaging for fast-slow mixed SPDEs driven by multiplicative fractional noise. The main goal of this part is to generalize the results in [@Hairer2019averaging; @pei2020averaging; @pei2020convergence; @peistochastic2017] by using directly a pathwise approach to deal with multiplicative fractional noise term. 