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The goal of this study is to check which, if any, of the published versionsof the pion and kaon fragmentation functions is compatible with the STAR dataon semi-inclusive pion and kaon production in proton-proton collisions, and onthe basis of this analysis to make reliable predictions for the $p_T$ spectraof the pions and kaons in inclusive pion and kaon production at the future NICAproton-proton collider. The calculations are carried out in next-to-leadingorder (NLO) of perturbative quantum chromodynamics (pQCD), using the welltested CTEQ6 parton distributions. We consider the following pion and kaonfragmentation functions (FFs) -- DSEHS-14, DSEHS-17, LSS-15, HKNS-07 andAKK-08. Our analysis shows that within the experimental errors all tested setsof fragmentation functions provide a good fit to STAR data at the c.m. energy $\sqrt{S} = 200\, {\rm GeV} $, and the best ones are both LSS-15 and DSEHS-14for pions and DSEHS-17 for kaons. From comparison of the LO and NLO results itis clear that the latter fit data much better, specially in the region of small$p_T$. The NLO cross sections are also less scale-$Q^2$ dependent, where$p_T/2\leqslant Q \leqslant 2p_T$, than the LO ones. In order to makepredictions for NICA energies, we compare the NLO pQCD results with theexisting experimental BES STAR data on semi-inclusive hadron production in themost peripheral Au+Au collisions where the nuclear effects can be neglected.The comparison for lower energy scales, like at NICA, shows that a purely pQCDapproach is inadequate and suggests the necessity to take into account alsohigher-order effects of initial-state soft-gluon radiation. Nevertheless, thesedata on the $p_T$ spectra of $\pi^+$, $K^+$ and also the ratios $\pi^-/\pi^+$and $K^-/K^+$ seem favour LSS-15 and DSEHS-14 FFs for pions and DSEHS-17 forkaons, similarly as at the energy scale $ \sqrt{S} = 200\, {\rm GeV} $.

Tests of the parametrizations of Fragmentation Functions using data on  inclusive pion and kaon production in unpolarized $pp$ collisions from the  STAR collaboration and at the NICA project