We state and prove a generalization of Kingman's ergodic theorem on a measure-preserving dynamical system $(X,\mathcal{F},μ,T)$ where the $μ$-almost sure subadditivity condition $f_{n+m} \leq f_n + f_m \circ T^{n}$ is relaxed to a $μ$-almost sure, "gapped", almost subadditivity condition of the form $f_{n+σ_m+m} \leq f_n +ρ_n + f_m \circ T^{n+σ_n}$ for some nonnegative $ρ_n \in L^1(\mathrm{d}μ)$ and $σ_n \in \mathbf{N} \cup \{0\}$ that are suitably sublinear in $n$. This generalization has a first application to the existence of specific relative entropies for suitably decoupled measures on one-sided shifts.