We study the SYK model with an extra constant source, i.e. a constant matrix or equivalently a diagonal matrix with only one non-zero entry $λ_1$. By using methods from analytic combinatorics, we find exact expressions for the moments of this model. We further prove that the spectrum of this model can have a gap when $λ_1>λ^c_1$, thus exhibiting a phase transition in $λ_1$. In this case, a single isolated eigenvalue splits off from SYK's eigenvalues distribution. We located this single eigenvalue by analyzing the singular behavior of a supercritical functional composition scheme. In certain limit our results recover the ones of random matrices with non-zero mean entries.