Rates of convergence with asymptotically negligibly noisy observations

Apologies in advance if this question is not completely well defined. Suppose that I am estimating a nonparametric model for a conditional expectation function $$\mathbb E[Y_i | X_i]$$ using some i.i.d. dataset. Suppose, however, that I do not directly observe $$Y_i$$, but instead, for each sample size, $$n$$, observe $$\psi_{i,n} = Y_i + o_P(1)$$, i.e. I observe $$Y_i$$ with noise, but the degree of corruption in each observation converges to 0 when I have a large sample. Let $$\hat g_n$$ be some abstract estimation procedure which takes $$n$$ observations of $$(X_i,Y_i$$ and spits out an estimator for the conditional expectation function of $$Y_i$$ given $$X_i$$. Moreover, suppose that given the population distribution for $$(X_i,Y_i)$$, $$\hat g_n$$ satisfies some rate condition, say, $$r_n(\hat g_n(x) - \mathbb E[Y_i | X_i = x]) \overset p\to 0$$ for all $$x$$ in the support of the $$X_i$$.

I an interested in conditions under which $$\hat g_n$$ converges to $$\mathbb E[Y_i | X_i = x]$$ if we plugged in $$\psi_{i,n}$$ into our estimator $$\hat g_n$$ instead of $$Y_i$$. In particular, does the $$o_P(1)$$ assumption on the error suffice, or do we need stronger conditions on the discrepancy between $$\psi_{i,n}$$ and $$Y_i$$ to get results. I'm guessing, for instance, noise that is $$o_P(q)$$ but is not even integrable can be highly problematic if $$\hat g_n$$ does not have a principled way to deal with outliers. Is there any literature I could look at that addresses questions like this?