Under ZF + AD, and especially $\text{AD}^+$, even if uniformization fails for reals, in some ways it must almost hold.
For a notion of small, we say that uniformization holds on a co-small set of reals iff $∀A⊆ℝ^2 \, ∃f \; \{x: ∃y \, A(x,y) ∧ ¬A(x,f(x))\} \text{ is small}$. We require smallness to be preserved under subsets, but that does not affect the question.
Does $\text{AD}^+$ prove that for every $r∈ℝ$ and $Σ^2_1(r)$ notion of small such that $∃z∈ℝ$ “ $\{x∈ℝ: z \text{ is } Δ^2_1(r,x)\}$ is small”, uniformization holds on a co-small set of reals?
I also have a second question on the theme of uniformization almost holding under $\text{AD}^+$:
Under $\text{AD}^+$, do we have uniformization for $A⊆ℝ^2$ such that for some prewellordering $≺$, $A[x]$ depends only on the $≺$-rank of $x$?
Note: My guess is that the answer is yes under $\text{AD}^+$ but not under just AD.
Other ways uniformization almost holds under $\text{AD}^+$ include uniformization for $A⊆ℝ^2$ if $A$ is $\mathbf{Σ^2_1}$ or $Θ^{\operatorname{HOD}(A,ℝ)} < Θ$ or $∀x \, |A[x]|≤ω$.
For the first question, the answer is yes (with just AD and Borel $f$) if small is 'null' (measure 0) or 'meager'; for example see here. Uniformization on a co-null or a co-meager set of reals (with a Borel $f$) also holds in the Solovay model (Theorem 1(5) in A model of set-theory in which every set of reals is Lebesgue measurable by Solovay).
Here is an inner model theory based proof for $L(ℝ)$ if small is 'null' or 'meager'; the argument generalizes to some other notions of small. Let $r$ be a real such that $A$ is ordinal definable in $L(ℝ)$ from $r$. Assume $ℝ^\#$ exists. The symmetric generic collapse (from up to the limit of the Woodin cardinals to $ω$) of $M_ω(r)$ is elementarily embeddable into $L(ℝ)$, with $A$ in the range of the embedding. Every $M_ω(r)$-random (resp. $M_ω(r)$-generic) real is inside such a collapse for some choice of the generic, and the set of such reals is co-null (resp. comeager). Using $M_ω^\#(r)$, given any real $x$ that is inside some such collapse of $M_ω(r)$, we can (definably; we get Borel $f$) choose and well-order such a collapse containing $x$, and then return the least $y∈A[x]$ under the well-ordering (assuming $∃y ∈ A(x,y)$). Finally, $ℝ^\#$ is dispensable by letting the mouse operator depend on $A$ instead of using $M_ω^\#$ (and using elementary embeddings into some $L_δ(ℝ)$).
For more complicated notions of small, one may need to iterate $M_ω(r)$ (or another mouse) to get genericity, and $f$ need no longer be Borel, but the conditions in the question plausibly ensure that the needed iterations are simple enough to be handled inside the model.
