  We investigate the Gross-Prasad conjecture and its refinement for the Bessel
periods in the case of
$\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$. In
particular, by combining several theta correspondences, we prove the
Ichino-Ikeda type formula for any tempered irreducible cuspidal automorphic
representations. As a corollary of our formula, we prove an explicit formula
relating certain weighted averages of Fourier coefficients of holomorphic
Siegel cusp forms of degree two which are Hecke eigenforms to central special
values of $L$-functions. The formula is regarded as a natural generalization of
the B\"ocherer conjecture to the non-trivial toroidal character case.
