  In the latest data release from the Fermi $\gamma$-Ray Space Telescope (the
4th Fermi LAT 12-year Catalog or 4FGL) more than 50% of the Galactic sources
are yet to be identified. We observed thirteen unidentified Fermi LAT sources
with Chandra X-Ray Observatory (CXO) to explore their nature. We report the
results of the classification of X-ray sources in the fields of these
$\gamma$-ray sources and discuss the implications for their nature. We use the
multiwavelength (MW) data for machine-learning classification accompanied by a
more detailed spectral/variability analysis for brighter sources. Seven 4FGL
sources have $\gamma$-ray pulsars within their position error ellipses. Three
of these pulsars are either detected in the CXO images or show hints of X-ray
emission. Within the positional uncertainties of three 4FGL sources we detect
X-ray sources that may be yet unknown pulsars, depending on the MW association.
In addition to point sources, we discovered 2 extended sources one of which is
likely to be a bowshock pulsar-wind nebula associated with PSR J1358.3-6026.
Finally, we classify other X-ray sources detected in these observations and
report most interesting classifications.
