Most massive stars belong to multiple systems, yet the formation process leading to such high multiplicity remain insufficiently understood. To help constrain the different formation scenarios that exist, insights on the low-mass end of the companion mass function of such stars is crucial. However, this is a challenging endeavour as (sub-)solar mass companions at angular separations {rho} below 1" (corresponding to 1000-3000au in nearby young open clusters and OB associations) are difficult to detect due to the large brightness contrast with the central star. With the Carina High-contrast Imaging Project of massive Stars (CHIPS), we aim to obtain statistically significant constraints on the presence and properties of low-mass companions around massive stars at a previously unreachable observing window ({Delta}mag>=10 at {rho}<= 1"). In this second paper in the series, we focus on the Trumpler 14 cluster, which harbours some of the youngest and most massive O-type stars in the Milky Way. We obtained VLT-SPHERE observations of seven O-type objects in Trumpler 14 using the IRDIFS_EXT mode. These allow us to search for companions at separations larger than 0.15" (~360au) and down to magnitude contrast >10 mag in the near-infrared. We used angular and spectral differential imaging along with PSF fitting to detect sources and measure their flux relative to that of the central object. We detected 211 sources with near-infrared magnitude contrast in the range of 2 to 12. The closest companion, at only 0.26", is characterised as a 1.4M_{sun}_ stars with an age of 0.6Myr, in excellent agreement with previous age estimates for Tr14. The mass function peaks at about 0.4M_{sun}_ and presents a dearth of stars in the 0.5 to 0.8M_{sun}_ mass range compared to previous estimates of the initial mass function in Tr14.