  A planar graph is essentially $4$-connected if it is 3-connected and every of
its 3-separators is the neighborhood of a single vertex. Jackson and Wormald
proved that every essentially 4-connected planar graph $G$ on $n$ vertices
contains a cycle of length at least $\frac{2n+4}{5}$, and this result has
recently been improved multiple times.
  In this paper, we prove that every essentially 4-connected planar graph $G$
on $n$ vertices contains a cycle of length at least $\frac{5}{8}(n+2)$. This
improves the previously best-known lower bound $\frac{3}{5}(n+2)$.
