In order to reach this objective, we shall borrow the construction of stochastic integral with respect to infinite-dimensional fBm given in [@Garrido-Atienzalu2010; @maslowski2003evolution] and the stopping time technique to control the fBm term given in [@Mishura2012mixed] which will be recalled in Section \[se-2\] and Section \[se-3\], respectively (See Theorem \[mainthm\]).

The paper is organized as follows. Section \[se-2\] presents some necessary notations and assumptions. Section \[se-3\] presents pathwise unique solutions for a class of mixed SPDEs driven by fBm and Bm. An averaging principle for fast-slow mixed SPDEs driven by fBm subject to an additional fast-varying diffusion process is then established in Section \[se-4\]. Appendix **A** provides the arguments of the ergodicity for the fast component in which the slow component is kept frozen. Some technical complements are included in Appendix **B**.

Preliminaries {#se-2}
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Although the results on fractional calculus and stochastic integrals with respect to the one-dimensional fBm $\beta^H$ and $V$-valued infinite-dimensional fBm $B^H$ have already been done in the recent paper [@Biagini2008stochastic; @GarridoAtienza2010dcds; @Garrido-Atienzalu2010; @maslowski2003evolution; @mishura2008stochastic], we present them here for the sake of completeness.

For $T>0$, let $W^{\alpha,1}(0,T;V)$ be the space of measurable functions $h:[0,T]\to V$ such that $$\begin{aligned}
    |h|_{\alpha,1}:=\int_0^T\Big(\frac{|{h(s)}|}{s^\alpha}+\int_0^s\frac{|{h(s)-h(r)}|}{(s-r)^{\alpha+1}}\mathrm{d}r\Big)\mathrm{d}s<\infty,\end{aligned}$$ where $0<\alpha<\frac{1}{2}$ is fixed. Following Zähle 