Let $X$ be a normal affine $\mathbb{T}$-variety of complexity at most one over a perfect field $k$, where $\mathbb{T} = \mathbb{G}_{\rm m}^{n}$ stands for the split algebraic torus. Our main result is a classification of additive group actions on $X$ that are normalized by the $\mathbb{T}$-action. This generalizes the classification given by the second author in the particular case where $k$ is algebraically closed and of characteristic zero. With the assumption that the characteristic of $k$ is positive, we introduce the notion of rationally homogeneous locally finite iterative higher derivations which corresponds geometrically to additive group actions on affine $\mathbb{T}$-varieties normalized up to a Frobenius map. As a preliminary result, we provide a complete description of these $\mathbb{G}_{a}$-actions in the toric situation.