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We study the Fourier coefficients of functions satisfying a certain versionof Wright's circle method with finitely many major arcs. We show that theJensen polynomials associated with such Fourier coefficients are asymptoticallyhyperbolic, building on the framework of Griffin--Ono--Rolen--Zagier andothers. Consequently, we prove that the Fourier coefficients asymptoticallysatisfy all higher-order Tur\'an inequalities. As an application, we apply ourresults to both $(q^t;q^t)_\infty^{-r}$, which counts $r$-coloured partitionsinto parts divisible by $t$, and to the function $(q^{a};q^{p})_\infty^{-1}$where $p$ is prime and $0\leq a<p$, a ubiquitous function throughout numbertheory.

Jensen polynomials associated with Wright's circle method: Hyperbolicity and Tur\'an inequalities