  AIMS: HESS J1844-030 is a newly confirmed TeV source in the direction of the
X-ray pulsar wind nebula (PWN) candidate G29.4+0.1 and the complex radio source
G29.37+0.1, which is likely formed by the superposition of a background radio
galaxy and a Galactic supernova remnant (SNR). We investigate the possible
connection between the SNR, the PWN G29.4+0.1, and HESS J1844-030 to shed light
on the astrophysical origin of the TeV emission. METHODS: We performed an
imaging and spectral study of the X-ray emission from the PWN G29.4+0.1 using
archival observations obtained with the Chandra and XMM-Newton telescopes.
Public radio continuum and HI data were used to derive distance constraints for
the SNR that is linked to G29.37+0.1 and to investigate the interstellar medium
where it is expanding. We applied a simple model of the evolution of a PWN
inside an SNR to analyze the association between G29.4+0.1 and the radio
emission from G29.37+0.1. We compared the spectral properties of the system
with the population of TeV PWNe to investigate if HESS J1844-030 is the very
high energy counterpart of the X-ray PWN G29.4+0.1. RESULTS: We conclude that
G29.4+0.1 is a PWN and that a point source embedded on it is the powering
pulsar. The HI data revealed that the SNR linked to G29.37+0.1 is a Galactic
source at 6.5 kpc and expanding in a nonuniform medium. From the analysis of
the pulsar motion and the pressure balance at the boundary of X-ray emission,
we conclude that G29.4+0.1 could be a PWN that is located inside its host
remnant, forming a new composite SNR. Based on the magnetic field of the PWN
obtained from the X-ray luminosity, we found that the population of electrons
producing synchrotron radiation in the keV band can also produce IC photons in
the TeV band. This suggests that HESS J1844-030 could be the very high energy
counterpart of G29.4+0.1.
