Let $W^H$ be a fBm with Hurst parameter $H$ and let $\mathcal{H}$ be its Cameron-Martin space. Then by Girsanov theorem we know that if $\mathbb{P}$ is an fBm measure, it holds that there exists a measure $\mathbb{Q}$, equivalent to $\mathbb{P}$, such that for every $h \in \mathcal{H}$ we have that $t \mapsto W^H_t + h_t$ is a fBm under $\mathbb{Q}$.
Then by AMN01, Theorem 1 for instance we know that the following formula holds:
$$f(W^H)_{st} = \int_s^t f'(W^H_r) \delta W^H_r + \int_s^t f''(W^H_r) r^{2H-1} dr $$
Does the same hold for $X = W^H + h$, i.e.
\begin{equation} f(X)_{st} = \int_s^t f'(X) \delta W^H_r + \int_s^t f'(X) dh_r + \int_s^t f''(X_r) r^{2H-1} dr \end{equation}
assuming that the second integral on the rhs can be well-defined, for instance $h \in \mathcal{C}_t^{\alpha}$ such that $H + \alpha > 1$, so that it can be understood as a Young integral.
My idea would be to use the first equation with $X$ instead of $W^H$. Integral is defined application of divergence operator on path $f'(X)$
$$f'(X) \in \text{Dom}\ \delta^{X}$$
by chapter 2 in the same paper we'd have that for $u$ piecewise constant:
$$\delta^X(u) = \sum_j u_j (X_{t_{j+1}} - X_{t_j}) = \sum_j u_j (W^H_{t_{j+1}} - W^H_{t_j}) + \sum_j u_j h_{t_j t_{j+1}}$$
so I'd like it to converge to what I want upon taking small partitions. In other words, my question can be boiled down to the following:
question is it true that for $X = W^H + h$ as above, we can write Skorohod integral with respect to $X$ as:
$$\delta^X(u) = \int_0^t u \delta X = \int_0^t u \delta (W^H + h) = \int_0^t u \delta W^H + \int_0^t u dh$$
we can assume additional continuity on $u$, such that the last integral makes sense as the Young one. Since later I'd probably like to keep Gaussianity of $X$, I'd like it to hold under measure $\mathbb{Q}$.
If it's not true, then it would mean that we have to show (I think, by Theorem 1 in the mentioned paper):
$$\mathbb{E} \int_0^t D_r G K^* [ f'(W^H_r + h_r) ] dr = \mathbb{E} [ G f(X)_{st} ] - \mathbb{E}[ G \int_s^t f'(X_r) dh_r ] - \frac{1}{2} \mathbb{E} \int_s^t f''(X_r) r^{2H-1} dr$$
for any $G \in L^2(\Omega)$ for $\mathbb{E}$ being expectation wrt $\mathbb{P}$. But $X$ is not necessarily Gaussian anymore (and certainly doesn't have zero mean under a measure $\mathbb{P}$), so that we would have to perform a change of measure which may make computations very painful.
note - I know that change of variables can be defined with help of rough paths theory, if we lift $W^H$ to the rough path. I'm specifically curious about Skorohod integral approach.
