The Spin Physics Detector at the future NICA collider aims for investigate the nucleon spin structure in collisions of longitudinally and transversely polarized protons and deuterons at $\sqrt{s}$ up to 27 GeV and luminosity up to 10$^{32}$ cm$^{-2}$ s$^{-1}$. It will operate as a universal facility for comprehensive study of unpolarized and polarized gluon content in the proton and deuteron. Such complementing probes as charmonia, open charm and prompt-photon production processes will be used for that. Possible physics tasks such as the access to the gluon helicity, gluon Sivers function and gluon transversity in the deuteron via the measurement of single and double spin asymmetries and other gluon-related tasks will be discussed.