We propose methods for Bayesian inference for a new class of semiparametric survival models with a cure fraction. Specifically, we propose a semiparametric cure rate model with a smoothing parameter that controls the degree of parametricity in the right tail of the survival distribution. We show that such a parameter is crucial for these kinds of models and can have an impact on the posterior estimates. Several novel properties of the proposed model are derived. In addition, we propose a class of improper noninformative priors based on this model and examine the properties of the implied posterior. Also, a class of informative priors based on historical data is proposed and its theoretical properties are investigated. A case study involving a melanoma clinical trial is discussed in detail to demonstrate the proposed methodology.