  We present Swift follow-up observations of a sample of 35 unclassified faint
X-ray sources drawn from the ASCA Galactic centre and plane surveys. Our short
pointed XRT observations allow detections down to a limiting 0.3-10 keV flux of
~1E-13 erg cm-2 s-1, which translates into a luminosity of ~1E33 erg s-1 for an
assumed distance of D=8 kpc. The brightest source in our sample reaches a
maximum 0.3-10 keV luminosity of ~2E36 (D/8kpc)^2 erg s-1 during our
observations. We detect 16 (46%) of the ASCA sources with the XRT, while 19
were not detected during our program. Since we are probing the faint end of the
ASCA source populations, we expect a large fraction of the non-detections to be
due to the Eddington bias. This is strengthened by the fact that we find the
observed XRT count rates to be predominantly lower than expected based on the
reported ASCA intensities. Investigation of the ASCA properties and long-term
X-ray variability leads us to conclude that the non-detections likely include
two spurious ASCA detections and three objects that are variable or transient
X-ray sources. For the 16 XRT-detected sources we obtain positional accuracies
of ~2-4", which significantly improves upon their ASCA uncertainties of ~1'. We
use the X-ray spectra and variability to characterise these objects. Most
appear to be faint, persistent X-ray emitters that have highly absorbed
spectra. Based on their X-ray properties we identify three accreting compact
objects: one confirmed (AX J1754.2-2754) and one candidate (AX J1538.3-5541)
X-ray binary, and one possible magnetically accreting white dwarf (AX
J1620.1-5002). Furthermore, we use the improved positions of XRT-detected
sources to search for counterparts in simultaneously obtained Swift/UVOT images
and possible associations with cataloged sources at various wavelengths. This
reveals three possible main sequence stars amongst our sample.
