There is a good discussion here: Zero-Inflated models in Stan - General - The Stan Forums
A mixture between continuous and discrete is not really a mixture and more a model with two outcomes/ likelihoods (one binomial for the discrete zeros and one continuous for the rest). Since there is no crosstalk between the two components one can model them separately or ignore one altogether (e.g drop the zeros) without loss of information for the kept parameters.
One exception is if there is considerable rounding going on that the (now) discretized continuous distribution can actually generate zeros.