We study the Fourier coefficients of functions satisfying a certain version of Wright's circle method with finitely many major arcs. We show that the Jensen polynomials associated with such Fourier coefficients are asymptotically hyperbolic, building on the framework of Griffin-Ono-Rolen-Zagier and others. Consequently, we prove that the Fourier coefficients asymptotically satisfy all higher-order Turán inequalities. As an application, we apply our results to both (q t ; q t ) −r ∞ , which counts r-coloured partitions into parts divisible by t, and to the function (q a ; q p ) −1 ∞ where p is prime and 0 ≤ a < p, a ubiquitous function throughout number theory.