  We give explicit pullback formulae for nearly holomorphic Saito-Kurokawa
lifts restrict to product of upper half-plane against with product of elliptic
modular forms. We generalize the formula of Ichino to modular forms of higher
level and free the restriction on weights. The explicit formulae provide
non-trivial examples for the refined Gan-Gross-Prasad conjecture for $({\rm
SO}_5,{\rm SO}_4)$ in the non-tempered cases. As an application, we obtain
Deligne's conjecture for critical values of certain automorphic $L$-functions
for ${\rm GL}_3 \times {\rm GL}_2$. We also expect to apply our pullback
formulae to construct two-variables $p$-adic $L$-functions for ${\rm GL}_3
\times {\rm GL}_2$ in the future.
