We characterized the transiting planetary system orbiting the G2V star K2-38 using the new-generation echelle spectrograph ESPRESSO. We carried out a photometric analysis of the available K2 photometric light curve of this star to measure the radius of its two known planets. Using 43 ESPRESSO high-precision radial velocity measurements taken over the course of 8 months along with the 14 previously published HIRES RV measurements, we modeled the orbits of the two planets through a MCMC analysis, significantly improving their mass measurements. Using ESPRESSO spectra, we derived the stellar parameters, $T_{rm eff}$=5731$pm$66, $log g$=4.38$pm$0.11~dex, and $[Fe/H]$=0.26$pm$0.05~dex, and thus the mass and radius of K2-38, $M_{star}$=1.03 $^{+0.04}_{-0.02}$~M$_{oplus}$ and $R_{star}$=1.06 $^{+0.09}_{-0.06}$~R$_{oplus}$. We determine new values for the planetary properties of both planets. We characterize K2-38b as a super-Earth with $R_{rm P}$=1.54$pm$0.14~R$_{rm oplus}$ and $M_{rm p}$=7.3$^{+1.1}_{-1.0}$~M$_{oplus}$, and K2-38c as a sub-Neptune with $R_{rm P}$=2.29$pm$0.26~R$_{rm oplus}$ and $M_{rm p}$=8.3$^{+1.3}_{-1.3}$~M$_{oplus}$. We derived a mean density of $rho_{rm p}$=11.0$^{+4.1}_{-2.8}$~g cm$^{-3}$ for K2-38b and $rho_{rm p}$=3.8$^{+1.8}_{-1.1}$~g~cm$^{-3}$ for K2-38c, confirming K2-38b as one of the densest planets known to date. The best description for the composition of K2-38b comes from an iron-rich Mercury-like model, while K2-38c is better described by a rocky model with a H2 envelope. The maximum collision stripping boundary shows how giant impacts could be the cause for the high density of K2-38b. The irradiation received by each planet places them on opposite sides of the radius valley. We find evidence of a long-period signal in the radial velocity time-series whose origin could be linked to a 0.25-3~M$_{rm J}$ planet or stellar activity.
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