  Let $f$ be a modular form which is non-ordinary at $p$. Kim and Loeffler have
recently constructed two-variable $p$-adic $L$-functions associated to $f$. In
the case where $a_p=0$, they showed that, as in the one-variable case,
Pollack's plus and minus splitting applies to these new objects. In this short
note, we show that such a splitting can be generalised to the case where
$a_p\ne0$ using Sprung's logarithmic matrix.
