  Nonreciprocity, which violates Lorentz reciprocity, plays a pivotal role in
quantum information processing and networks. Nevertheless, achieving a desired
and highly-tunable level of nonreciprocity has proven to be a formidable
challenge. Here, we propose a coupled nonlinear cavity magnonic system,
consisting of two cavities, a second-order nonlinear element, and a
yttrium-iron-garnet sphere for supporting Kerr magnons, to realize this
sought-after highly-tunable nonreciprocity. We first derive the critical
condition for switching between reciprocity and nonreciprocity with undriven
magnons, then we numerically demonstrate that a strong magnonic nonreciprocity
can be obtained by breaking the critical condition. When magnons are driven, we
show that a strong magnonic nonreciprocity can also be attained within the
critical condition. Compared to previous study, the introduced nonlinear
element not only relaxes the critical condition to both the weak and strong
regimes, but also offers an alternative path to adjust the magnonic
nonreciprocity. Our work provides a promising avenue to realize highly-tunable
nonreciprocal devices with Kerr magnons.
