  Variable-length compression without prefix-free constraints and with
side-information available at both encoder and decoder is considered. Instead
of requiring the code to be error-free, we allow for it to have a non-vanishing
error probability. We derive one-shot bounds on the optimal average codeword
length by proposing two new information quantities; namely, the conditional and
unconditional $\varepsilon$-cutoff entropies. Using these one-shot bounds, we
obtain the second-order asymptotics of the problem under two different
formalisms---the average and maximum probabilities of error over the
realization of the side-information. While the first-order terms in the
asymptotic expansions for both formalisms are identical, we find that the
source dispersion under the average error formalism is, in most cases, strictly
smaller than its maximum error counterpart. Applications to a certain class of
guessing problems, previously studied by Kuzuoka [\emph{{IEEE} Trans.\ Inf.\
Theory}, vol.~66, no.~3, pp.~1674--1690, 2020], are also discussed.
