In analysis of time-to-event data, the mixture cure (MC) model is more appropriate than conventional Cox proportional hazards model when the study sample includes long-term survivors. However, in samples with few events, standard maximum likelihood estimation (MLE) can be biased. We extend Firth-type penalized likelihood estimation (FT-PLE) developed for bias reduction in the exponential family to the Weibull-logistic MC, using the Jeffreys invariant prior. Via simulation studies based on a motivating analysis of multiple protein biomarkers for disease recurrence in breast cancer, we compare FT-PLE vs MLE for effect estimate bias, and type 1 error and power for likelihood ratio statistics(LR), including log-F prior and modified FT-PLE methods. Under a low event-rate model, FT-PLE estimates could always be obtained when MLEs were infinite, and had smallest mean bias among all four methods. We demonstrate good type 1 error control for FT-PLE LR and better power than MLE LR.