Given a pair (semispray $S$, almost symplectic form $\omega$) on a tangent bundle, the family of nonlinear connections $N$ such that $\omega$ is recurrent with respect to $(S,N)$ with a fixed recurrent factor is determined by using the Obata tensors. In particular, we obtain a characterization for a pair $(N,\omega)$ to be recurrent as well as for the triple $(S,\overset{c}{N},\omega)$ where $\overset{c}{N}$ is the canonical nonlinear connection of the semispray $S$. In the particular case of vanishing recurrence factor we get the family of almost Fedosov structures associated to a fixed semispray and almost symplectic structure. For a triple (semispray $S$, almost symplectic form $\omega$, metric $g$), a characterization for existence of a corresponding almost metriplectic structure is obtained.