$k$-contraction is a generalization of the classical contraction property. It allows the study of complex behaviors in partially stable systems. However, existing conditions for kcontraction are often intractable. This work proposes efficiently solvable sufficient conditions for k-contraction verification in partially linear systems. Our findings are derived by exploiting particular quadratic abstractions arising from classical Lur’e systems analysis. We specialize our result to nonlinearities satisfying shifted monotonicity and differential sector-bound properties. We showcase the potential of our method by designing nonlinear controllers for linear systems, achieving complex closed-loop behaviors.