  Very-high-energy (VHE) observations have revealed approximately 100 TeV
sources in our Galaxy, and a significant fraction of them are under
investigation for understanding their origin. We report our study of one of
them, HESS~J1844$-$030. It is found possibly associated with the supernova
remnant (SNR) candidate G29.37+0.1, and detailed studies of the source region
at radio and X-ray frequencies have suggested that this SNR is a composite one,
containing a pulsar wind nebula (PWN) powered by a candidate young pulsar. As
the GeV source 4FGL~J1844.4$-$0306 is also located in the region with high
positional coincidence, we analyze its $\gamma$-ray data obtained with the
Large Area Telescope on-board the {\it Fermi Gamma-ray Space Telescope }. We
determine the GeV $\gamma$-ray emission is extended, described with a
Log-Parabola function. The obtained spectrum can be connected to that of the
VHE source HESS J1844$-$030. Given the properties and those from
multi-frequency studies, we discuss the origin of the $\gamma$-ray emission by
considering that the two \gr\ sources are associated. Our modeling indicates
that while the TeV part would have either a hadronic (from the SNR) or a
leptonic origin (from the putative PWN), the GeV part would arise from a
hadronic process. Thus we conclude that 4FGL~J1844.4$-$0306 is the likely GeV
counterpart to G29.37+0.1.
