Half classification, half regression model? I'm trying to implement a model that predicts whether a user pays for something, and if so, how long it takes for them to pay. I intuit the output of this being either a label (0) or a real number. Is there such thing as a half classification, half regression model? Or am I misinterpreting the problem?
Edit: to clarify, paying or not paying in a certain time frame makes a difference, with those not paying effectively having a False/0 label. The more detailed context is that this is a prepayment model. (I see how my original explanation may be confusing).

You actually have two problems here, and you can treat them like that.

whether a user pays for something

You probably want to model a Bernoulli trial here.
You'll have a probability $p(X=x) \in(0,1)$.
So you have a loss function $\mathcal L_B(y,\hat y)=-y\log(\hat y) - (1-y)\log(1-\hat y)$, where $\hat y = f(x)$ is the probability that the user paid for something.

and if so, how long it takes for them to pay

This one results in a time $\tau(X=x) \in (0,+ \infty)$.
You can model this as an exponential, that will lead us to the following loss function: $\mathcal L_E(t,\hat t)=t\hat t - \log\hat t$, where $\hat t = 1/g(x)$.
So, the total loss-function (for a datum) is:
$$\mathcal L = \mathcal L_B + y\mathcal L_E$$
You can use the actual $y$ in training, which implies that
$$\mathcal L=\begin{cases}\mathcal L_B, \quad\text{for} \quad y=0 \\\mathcal L_B + \mathcal L_E, \quad\text{for} \quad y=1\end{cases}$$
This basically means estimating two separate models, unless $g$ and $f$ share parameters (if we are talking about neural networks, that can be a sensible choice, depending on the nature of $x$).
