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We investigate the Gross-Prasad conjecture and its refinement for the Besselperiods in the case of$\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$. Inparticular, by combining several theta correspondences, we prove theIchino-Ikeda type formula for any tempered irreducible cuspidal automorphicrepresentations. As a corollary of our formula, we prove an explicit formularelating certain weighted averages of Fourier coefficients of holomorphicSiegel cusp forms of degree two which are Hecke eigenforms to central specialvalues of $L$-functions. The formula is regarded as a natural generalization ofB\"ocherer's conjecture to the non-trivial toroidal character case.

On the Gross-Prasad conjecture with its refinement for  $\left(\mathrm{SO}\left(5\right),\mathrm{SO}\left(2\right)\right)$ and the  generalized B\"ocherer conjecture