For $0\leq α< 1$, the $\mathcal{A}_α$-spectral radius of a $k$-uniform hypergraph $G$ is defined to be the spectral radius of the tensor $\mathcal{A}_α(G):=α\mathcal{D}(G)+(1-α) \mathcal{A}(G)$, where $\mathcal{D}(G)$ and $A(G)$ are diagonal and the adjacency tensors of $G$ respectively. This paper presents several lower bounds for the difference between the $\mathcal{A}_α$-spectral radius and an average degree $\frac{km}{n}$ for a connected $k$-uniform hypergraph with $n$ vertices and $m$ edges, which may be considered as the measures of irregularity of $G$. Moreover, two lower bounds on the $\mathcal{A}_α$-spectral radius are obtained in terms of the maximum and minimum degrees of a hypergraph.