Planets with radii between 2 and 4R_{Earth} closely orbiting solar-type stars, are crucial for studying the transition from rocky to giant planets. These are prime targets for atmospheric characterization using missions like JWST and ARIEL. Unfortunately, only a handful of such planets with precise mass measurements are known to orbit bright stars. Our goal is to determine the mass of a transiting planet around the very bright F6 star, HD 73344 (Vmag=6.9). This star exhibits high activity and has a rotation period close to the planet's orbital period (P_b=15.6-days). The transiting planet, initially a K2 candidate, is confirmed through TESS observations (TOI 5140.01). We refine its parameters using TESS data and rule out a false positive with Spitzer observations. We analyze high-precision radial velocity (RV) data from the SOPHIE and HIRES spectrographs. We conduct separate and joint analyses of K2, TESS, SOPHIE, and HIRES data using the PASTIS software. Given the star's early type and high activity, we use a novel observing strategy, targeting the star at high cadence for two consecutive nights with SOPHIE to understand short-term stellar variability. We model stellar noise with two Gaussian processes: one for rotationally modulated stellar processes, one for short-term stellar variability. High-cadence RV observations provide better constraints on stellar variability and precise orbital parameters for the transiting planet: a radius of R_b_=2.98_-0.07_^+0.08^R_{Earth} and a mass of M_b=2.98_-1.90_^+2.50^M_{Earth} (upper-limit at 3{sigma} is <10.48M{Earth}). The derived mean density suggests a sub-Neptune-type composition, but uncertainties in the planet's mass prevent a detailed characterization. In addition, we detect a planetary candidate with a minimum mass of about M_c_sini_c=116.3_-13.0_^+12.8^M_{Earth} and a period of P_c=66.45+-0.25_^+0.10^days in the RV data. Dynamical analyses confirm the stability of the two-planet system and provide constraints on the inclination of the candidate planet, which favours a near-coplanar system. While the transiting planet orbits a bright star at a short period, stellar activity has prevented us from precise mass measurements despite intensive RV follow-up. Long-term RV tracking of this planet could improve this measurement, as well as our understanding of the host star's activity. The latter will be essential if we are to characterize the atmosphere of planets around F-type stars using transmission spectroscopy.