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In contrast, for plurisubharmonic functions and their subextensions the situation is more difficult: In: Fornæss, John Erik; Sibony, Nessim: Plurisubharmonic functions on ring domains. Complex analysis (University Park, Pa., 1986), 111–120, Lecture Notes in Math., 1268, Springer, Berlin, 1987 (MR0907057), the authors consider plurisubharmonic (psh) function on $B\setminus K$, where $B$ is a ball in $\mathbb{C}^n$ and $K$ is a polynomially convex set. They give the following counterexamples: one is a smooth psh function on $B\setminus K$, $K$ a polydisk, which does not have a psh subextension to $B$; another one is a discontinuous psh function on $B\setminus K$, $K$ a smaller ball, which does not have a psh subextension to $B$. There is also earlier work about psh functions that cannot be extended across a pseudoconvex set: Bedford, Eric; Burns, Dan Domains of existence for plurisubharmonic functions. Math. Ann. 238 (1978), no. 1, 67–69 (MR0510308).