Let C be an additive category with an involution ∗. Suppose that ϕ : X→ X is a morphism of C with core inverse ϕ #© : X → X and η : X → X is a morphism of C such that 1X + ϕ #©η is invertible. Let α = (1X +ϕ #©η)−1, β = (1X +ηϕ #©)−1, ε = (1X−ϕϕ #©)ηα(1X−ϕ #©ϕ), γ = α(1X−ϕ #©ϕ)β−1ϕϕ #©β, σ = αϕ #©ϕα−1(1X−ϕϕ #©)β, δ = β∗(ϕ #©)∗η∗(1X − ϕϕ #©)β. Then f = ϕ + η − ε has a core inverse if and only if 1X − γ, 1X − σ and 1X − δ are invertible. Moreover, the expression of the core inverse of f is presented. Let R be a unital ∗-ring and J(R) its Jacobson radical, if a ∈ R #© with core inverse a #© and j ∈ J(R), then a + j ∈ R #© if and only if (1 − aa #©) j(1 + a #© j)−1(1 − a #©a) = 0. We also give the similar results for the dual core inverse,