In earlier work by Chen and Gasparini it was found that leading-order finite-size scaling for the specific heat of helium confined to films and pores did not seem to hold. We present here an analysis which includes correction-to-scaling terms. We do this in two ways: First, we use equations for the shift and value of the specific-heat maximum as suggested by Fisher. Secondly, we show that a bulk-plus-surface specific-heat model yields very similar equations for the shift and maximum, and in addition suggests a simple power law for the scaling function near Ti, . All aspects of the data can be fitted very simply with this model with only three parameters.We find, however, that the surface specific-heat exponent Ot, does not agree with the scaling prediction e, =a+ v. When correction-to-scaling terms are introduced to force agreement, we find these terms to be very large, larger than the leading terms, and several orders of magnitude larger than the equivalent correction-to-scaling amplitudes for bulk helium. We believe our analysis in terms of the surface specific heat is the first for an experimental system, and we find that it is quite in keeping with the spirit of some theoretical calculations.