When a study sample includes a large proportion of long-term survivors, mixture cure (MC) models that separately assess biomarker associations with long-term recurrence-free survival and time to disease recurrence are preferred to proportional hazards models. Standard maximum likelihood estimation (MLE) may be biased in small or sparse samples (i.e. with few recurrences). We extend Firth-type penalized likelihood estimation (PLE) developed for bias reduction in the exponential family to the Weibull-logistic MC, using the Jeffreys invariant prior. Via simulation studies, we evaluate PLE, as well as type 1 error and power obtained using Wald-type and likelihood ratio statistics, in comparison to MLE. In samples with few events, the MC-PLEs had mean bias closer to zero and smaller mean squared error than MC-MLEs, and could be obtained in samples when the MLEs are infinite. We illustrate the practical utility of the methodology in a breast cancer cohort with long-term follow-up.