The production cross-section of the $\chi c1(3872)$ state relative to the $\psi (2S)$ meson is measured using proton-proton collision data collected with the LHCb experiment at centre-of-mass energies of $s=8$ and 13 TeV, corresponding to integrated luminosities of 2.0 and 5.4 fb$\mathrm{<mrow\; data-mjx-texclass="ORD">-1</mrow>}$, respectively. The two mesons are reconstructed in the $J/\psi \pi +\pi -$ final state. The ratios of the prompt and nonprompt $\chi c1(3872)$ to $\psi (2S)$ production cross-sections are measured as a function of transverse momentum, $pT$, and rapidity, $y$, in the kinematic range GeV/$c$ and . The prompt ratio is found to increase with $pT$ independently of $y$. The double-differential cross-section ratio of $\chi c1(3872)$ relative to $\psi (2S)$ mesons between 13 and 8 TeV is also measured for the prompt component and observed to be consistent with unity, independent of $pT$ and centre-of-mass energy.