averaging principle in the mean square sense for a class of SPDEs driven by fBm subject to an additional fast-varying diffusion process, which is given by $$\begin{aligned}
\label{spde-fs}
\begin{split}
\begin{cases}
\mathrm{d} X^{\varepsilon}_t&=(AX^{\varepsilon}_t+b( X^{\varepsilon}_t,Y^{\varepsilon}_t))\mathrm{d}t+g(X^{\varepsilon}_t)\mathrm{d}B^H_t,\\ \mathrm{d} Y^{\varepsilon}_t&\disp =\frac{1}{\varepsilon}(AY^{\varepsilon}_t+F(X^{\varepsilon}_t,Y^{\varepsilon}_t))\mathrm{d}t+\frac{1}
{\sqrt{\varepsilon}}G(X^{\varepsilon}_t,Y^{\varepsilon}_t)\mathrm{d}W_t,
\end{cases}
\end{split}\end{aligned}$$ where $b,g,F,G$ are nonlinear coefficients and $X^{\varepsilon}_0=X_{0},Y^{\varepsilon}_0=Y_{0}$ are initial values. The parameter $0<\varepsilon \ll 1$ represents the ratio between the natural time scale of the $X_{t}^{\varepsilon}$ and $Y_{t}^{\varepsilon}$ variables. For more precise setting and assumptions, see Section 4.

The theory of stochastic averaging principles has been studied extensively (see for instance the paper by Khasminskii [@khas1966limit] and Freidlin and Wentzell [@freidlin2012random], or the recent paper by Xu et al. [@xu2011averaging; @xu2014; @xu2015approximation; @Xu2015Stochastic; @Xu2017Stochastic], Liu [@liu2010strong], Liu, Röckner, Sun and Xie [@Liu2019] and Thompson, Kuske, and Monahan [@thompson2015stochastic]) and is used in many applications. Cerrai and Freidlin [@cerrai2009averaging] developed stochastic averaging principles for two-time-scale stochastic reaction diffusion equations whose additive noise is included in the fast motion. In this infinite-dimensional setting, there are also interesting papers such as Bréhier [@brehier2012strong], Xu and Miao [@xu2015strong], Fu et al. [@fu2011strong; @fu2015strong], Bao, Yin, and Yuan [@Bao2016Two] and Sun and Zhai [@sun2020baveraging]. However, the literature concerning fast-slow mixed SPDEs driven by multiplicative fractional noise 