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Past work has proven the possibility of extracting the top-quark mass, one of the fundamental parameters of the Standard Model, from the comparison of theory predictions and experimental measurements of differential cross-sections for $t\bar{t} j~+~X$ hadroproduction. Various experimental analyses in this respect have already been performed, and new ones are in preparation on the basis of the latest data from $pp$ collisions collected at the Large Hadron Collider. We have produced and made public a comprehensive set of theoretical predictions for the relevant differential distributions, ready to be used for presently ongoing and forthcoming experimental analyses. We investigate the role of different theoretical inputs, in particular the factorization and renormalization scales, PDFs and top-quark mass renormalization schemes, and we quantify the uncertainties related to different choices for these inputs, providing recommendations.