We introduce the pinning model on a quenched renewal, which is an instance of a (strongly correlated) disordered pinning model. The potential takes value 1 at the renewal times of a quenched realization of a renewal process $σ$, and $0$ elsewhere, so nonzero potential values become sparse if the gaps in $σ$ have infinite mean. The "polymer" -- of length $σ_N$ -- is given by another renewal $τ$, whose law is modified by the Boltzmann weight $\exp(β\sum_{n=1}^N \mathbf{1}_{\{σ_n\inτ\}})$. Our assumption is that $τ$ and $σ$ have gap distributions with power-law-decay exponents $1+α$ and $1+\tilde α$ respectively, with $α\geq 0,\tilde α>0$. There is a localization phase transition: above a critical value $β_c$ the free energy is positive, meaning that $τ$ is \emph{pinned} on the quenched renewal $σ$. We consider the question of relevance of the disorder, that is to know when $β_c$ differs from its annealed counterpart $β_c^{\rm ann}$. We show that $β_c=β_c^{\rm ann}$ whenever $ α+\tilde α\geq 1$, and $β_c=0$ if and only if the renewal $τ\capσ$ is recurrent. On the other hand, we show $β_c>β_c^{\rm ann}$ when $ α+\frac32\, \tilde α<1$. We give evidence that this should in fact be true whenever $ α+\tilde α<1$, providing examples for all such $ α,\tilde α$ of distributions of $τ,σ$ for which $β_c>β_c^{\rm ann}$. We additionally consider two natural variants of the model: one in which the polymer and disorder are constrained to have equal numbers of renewals ($σ_N=τ_N$), and one in which the polymer length is $τ_N$ rather than $σ_N$. In both cases we show the critical point is the same as in the original model, at least when $ α>0$.

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