  Cocyclic Hadamard matrices (CHMs) were introduced by de Launey and Horadam as
a class of Hadamard matrices with interesting algebraic properties. \'O
Cath\'ain and R\"oder described a classification algorithm for CHMs of order
$4n$ based on relative difference sets in groups of order $8n$; this led to the
classification of all CHMs of order at most 36. Based on work of de Launey and
Flannery, we describe a classification algorithm for CHMs of order $4p$ with
$p$ a prime; we prove refined structure results and provide a classification
for $p \leqslant 13$. Our analysis shows that every CHM of order $4p$ with
$p\equiv 1\bmod 4$ is equivalent to a Hadamard matrix with one of five distinct
block structures, including Williamson type and (transposed) Ito matrices. If
$p\equiv 3 \bmod 4$, then every CHM of order $4p$ is equivalent to a Williamson
type or (transposed) Ito matrix.
