How to write unnormalized posterior when prior is a mixture of continuous and discrete

Suppose I want to do bayesian inference on the regression problem $\beta$ for Y = X$\beta$ + $\epsilon$ for $\epsilon_i \sim N(0,\sigma^2)$. The complication is that the prior for each component $\beta_i$ is a mixture of a point mass at 0 and a normal (I think this is called a spike and slab prior).

For example, $\beta_j \sim p\delta_{0} + (1-p)G$ for $G \sim N(0,\tau^2)$.

I want to write down the product of the prior and the likelihood but I find it difficult to do this with the dirac delta function in the prior. I tried introducing a latent variable to indicate which component of the prior I'm drawing from, but I wonder if there is a more principled way to write down the product of the likelihood and the prior.

I seem to have this problem generally whenever there is a dirac delta function in a likelihood or prior. Any general tips would be appreciated.

Thanks.