  This paper takes the first steps towards a systematic study of additive rigid
meromorphic cocycles of higher weight. These were introduced by Darmon and
Vonk, who focused on multiplicative and weight two cocycles. After classifying
certain rigid meromorphic cocycles of weight $2k$, we construct an explicit
holomorphic kernel function realising a Shimura-Shintani style correspondence
from modular forms of weight $k+1/2$ and level $4p^2$ to rigid analytic
cocycles of weight $2k$ on SL$_2(\mathbb{Z}[1/p])$.
