  Building on our previous work, we classify all planar $p$-elasticae under the
pinned boundary condition, and then obtain uniqueness and geometric properties
of global minimizers. As an application we establish a Li--Yau type inequality
for the $p$-bending energy, and in particular discover a unique exponent $p
\simeq 1.5728$ for full optimality. We also prove existence of minimal
$p$-elastic networks, extending a recent result of Dall'Acqua--Novaga--Pluda.
