We present a study of the properties of the star-cluster systems around pseudobulges of late-type spiral galaxies using a sample of 11 galaxies with distances from 17 Mpc to 37 Mpc. Star clusters are identified from multiband Hubble Space Telescope ACS and WFPC2 imaging data by combining detections in three bands (F435W and F814W with ACS and F606W with WFPC2). The photometric data are then compared to population synthesis models to infer the masses and ages of the star clusters. Photometric errors and completeness are estimated by means of artificial source Monte Carlo simulations. Dust extinction is estimated by considering F160W NICMOS observations of the central regions of the galaxies, augmenting our wavelength coverage. In all galaxies we identify star clusters with a wide range of ages, from young (age ≾ 8 Myr) blue clusters, with typical mass of 10^3 M_⊙ sun to older (age >100-250 Myr), more massive, red clusters. Some of the latter might likely evolve into objects similar to the Milky Way's globular clusters. We compute the specific frequencies for the older clusters with respect to the galaxy and bulge luminosities. Specific frequencies relative to the galaxy light appear consistent with the globular cluster specific frequencies of early-type spirals. We compare the specific frequencies relative to the bulge light with the globular cluster specific frequencies of dwarf galaxies, which have a surface brightness profile that is similar to that of the pseudobulges in our sample. The specific frequencies we derive for our sample galaxies are higher than those of the dwarf galaxies, supporting an evolutionary scenario in which some of the dwarf galaxies might be the remnants of harassed late-type spiral galaxies that hosted a pseudobulge.