In this paper, we obtain approximate bound-state solutions of N-dimensional time-independent fractional Schrödinger equation for the generalised pseudoharmonic potential which has the form V(rα) = a1r2 α+ (a2/ r2 α) + a3. Here α(0<α<1) acts like a fractional parameter for the space variable r. The entire study consists of the Jumarie-type fractional derivative and the elegance of Laplace transform. As a result, we can successfully express the approximate bound-state solution in terms of Mittag–Leffler function and fractionally defined confluent hypergeometric function. Our study may be treated as a generalisation of all previous works carried out on this topic when α= 1 and N arbitrary. We provide numerical result of energy eigenvalues and eigenfunctions for a typical diatomic molecule for different α close to unity. Finally, we try to correlate our work with a Cornell potential model which corresponds to α= 1 / 2 with a3= 0 and predicts the approximate mass spectra of quarkonia. © 2019, Indian Academy of Sciences.