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The production of 𝜂 and 𝜂′ mesons is studied in proton-proton and proton-lead collisions collected with the LHCb detector. Proton-proton collisions are studied at center-of-mass energies of 5.02 and 13TeV and proton-lead collisions are studied at a center-of-mass energy per nucleon of 8.16TeV. The studies are performed in center-of-mass (c.m.) rapidity regions 2.5<𝑦c.m.<3.5 (forward rapidity) and −4.0<𝑦c.m.<−3.0 (backward rapidity) defined relative to the proton beam direction. The 𝜂 and 𝜂′ production cross sections are measured differentially as a function of transverse momentum for 1.5<𝑝T<10GeV and 3<𝑝T<10GeV, respectively. The differential cross sections are used to calculate nuclear modification factors. The nuclear modification factors for 𝜂 and 𝜂′ mesons agree at both forward and backward rapidity, showing no significant evidence of mass dependence. The differential cross sections of 𝜂 mesons are also used to calculate 𝜂/𝜋0 cross-section ratios, which show evidence of a deviation from the world average. These studies offer new constraints on mass-dependent nuclear effects in heavy-ion collisions, as well as 𝜂 and 𝜂′ meson fragmentation.