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Context. 1LHAASO J1740+0948u is a very-high-energy (VHE) source initially reported in the first catalogue by the LHAASO Collaboration, with no previous identifications and no counterpart at other wavelengths. It is detected by the KM2A instrument only, i.e. at energies above 25 TeV, with a 17.1σ significance, and also above 100 TeV at a 9.4σ level. It is located (σ RA,Dec ~ 0.02° at 95% confidence) at 0.22° from PSR J1740+1000, a faint radio and gamma-ray pulsar placed well above the Galactic plane ( b = 20.4°) that displays a long X-ray tail. Despite the offset, the two sources are likely associated with each other, since no other object has been found nearby at such a high Galactic latitude. Aims. We aim to study the diffuse X-ray emission around PSR J1740+1000 and its tail-like pulsar wind nebula (PWN) with XMM-Newton to investigate the origin of 1LHAASO J1740+0948u through a multi-wavelength spectral energy distribution (SED) fitting, testing different scenarios. Methods. We analysed ~500 ks of XMM-Newton observations of PSR J1740+1000. We studied, for the first time, the diffuse emission in two different regions: one centred on the pulsar and the other located inside the 1LHAASO J1740+0948u source region. We also studied the X-ray tail and how its emission evolves as a function of the distance from the pulsar. We then performed a fit of the SED, including the spectrum of 1LHAASO J1740+0948u and the X-ray data obtained from either the analysis of the PWN or the diffuse emission, to understand whether one of the two X-ray sources could be related to the VHE emission and attempt a source classification. Results. The X-ray analysis of the diffuse emission resulted in upper limits in the range of 0.5-10 keV. The tail-like PWN is best fitted with an absorbed power law with Γ = 1.76 ± 0.06 in the 0.5-8 keV range, with no significant detection of spectral variations with distance. The SED modelling, assuming the VHE emission to be only due to the X-ray tail, constrains its magnetic field to B = 6.8 ± 1.9 μG, which is in line with previous results. However, we do not find a good fit that could explain both the X-rays of the tail and the LHAASO spectrum with reasonable parameters, hinting that the VHE emission likely comes from an older X-ray-faint electron population. We then performed a SED fitting of the VHE spectrum combined with the upper limits on the diffuse X-ray emission, constraining the magnetic field to be as low as B ≤ 1.2 μG. We suggest that 1LHAASO J1740+0948u could represent either the relic PWN of PSR J1740+1000 or its pulsar halo. Based on our best-fit results, we estimated the energy density and obtained values ranging from 0.03 to 0.67 eV/cm 3 , depending on the spectral index of the electron distribution. These very low values suggest a halo-like nature for 1LHAASO J1740+0948u, but deeper multi-wavelength observations are required to confirm this hypothesis.