In closed monoidal category we have $$ \text{Hom}(X \otimes Y, Z) \cong \text{Hom}(X,[Y,Z]) $$ This way we can curry a function of two arguments. In many categories, the tensor product and product(as a limit) coincide, especially in $\mathbf{Set}$ where both products are just a cartesian product.
However, those two products do not always coincide, for example in category of vector spaces. Therefore, we cannot curry addition. More generally, in many F-algebras we have an operation $X \times X \rightarrow X$, that cannot be curried if tensor and normal product are not the same.
However in the case of vector spaces, we can forget some structure and obtain affine spaces where the tensor product work a bit differently. In particular, there is a map $p: X\otimes Y \rightarrow X\times Y$ such that $i \circ p = \text{id}$, where $i$ is the injection $i: X\times Y \rightarrow X\otimes Y$.
If I got it right, this map $p$ really exist only for tensor product in category of affine spaces and does not exist in category of vector spaces.
By composing this $p$ with addition, we can curry addition. The result is an affine map but that is to be expected.
I would be interested if this relation between vector and affine spaces is an instance of a more general idea? Maybe isn't there some construction which would take a F-algebra, forget some structure and obtain a category where we can curry operations?(probably with the above construction with $p$)