  We develop a theory of $p$-adic N\'eron functions on abelian varieties,
depending on various auxiliary choices, and show that the global $p$-adic
height functions constructed by Mazur and Tate can be decomposed into a sum of
$p$-adic N\'eron functions if the same auxiliary choices are made. We also
decompose the $p$-adic height constructed by Coleman and Gross, and extended to
arbitrary reduction by Colmez and Besser, into a sum of local height functions
for Jacobians of odd degree genus $2$ curves. We show that this local height
function is equal to the $p$-adic N\'eron function with the same auxiliary
choices, regardless of the reduction type of the curve. This extends work of
Balakrishnan and Besser for elliptic curves. When the curve has semistable
reduction and the reduction of the Jacobian is ordinary, we also describe the
$p$-adic N\'eron function that arises from the canonical Mazur-Tate splitting
explicitly in terms of a generalisation of the $p$-adic sigma function
constructed by Blakestad.
