  This paper generalises the result of Jean-Pierre Demailly on his
Ohsawa--Takegoshi-type $L^2$ extension theorem, which guarantees holomorphic
extensions for some sections $f$ on analytic subspaces $Y$ defined by
multiplier ideal sheaves of plurisubharmonic functions. The result in this
paper provides $L^2$ estimates for the extended sections even if those $f$ do
not vanish on the singular sets of the reduced loci of the subspaces $Y$, at
least for the case when the ambient manifold $X$ is compact and the relevant
metrics have only neat analytic singularities. This is achieved by replacing
the generalised Ohsawa measure by a measure supported on log-canonical centres.
