New lattice data for the Pi_u and Sigma_u^- potentials at short distances are presented. We compare perturbation theory to the lower static hybrid potentials and find good agreement at short distances, once the renormalon ambiguities are accounted for. We use the non-perturbatively determined continuum-limit static hybrid and ground state potentials at short distances to determine the gluelump energies. The result is consistent with an estimate obtained from the gluelump data at finite lattice spacings. For the lightest gluelump, we obtain Lambda_B^{RS}(nu_f=2.5/r_0)=[2.25\pm 0.10(latt.)\pm 0.21 (th.)\pm 0.08 (Lambda_{MS})]/r_0 in the quenched approximation with r_0^{-1} approx 400 MeV. We show that, to quote sensible numbers for the absolute values of the gluelump energies, it is necessary to handle the singularities of the singlet and octet potentials in the Borel plane. We propose to subtract the renormalons of the short-distance matching coefficients, the potentials in this case. For the singlet potential the leading renormalon is already known and related to that of the pole mass, for the octet potential a new renormalon appears, which we approximately evaluate. We also apply our methods to heavy light mesons in the static limit and from the lattice simulations available in the literature we obtain the quenched result \bar{Lambda}^{RS}(nu_f=2.5/r_0)=[1.17\pm 0.08(latt.)\pm 0.13 (th.) \pm 0.09 (Lambda_{MS})]/r_0. We calculate m_{b,MS}(m_{b,MS}) and apply our methods to gluinonia whose dynamics are governed by the singlet potential between adjoint sources. We can exclude non-standard linear short-distance contributions to the static potentials, with good accuracy.