Aims. Previous observations with the HESS telescope array revealed the existence of extended very-high-energy (VHE; E > 100 GeV) γ-ray emission, HESS J1023-575, coincident with the young stellar cluster Westerlund 2. At the time of discovery, the origin of the observed emission was not unambiguously identified, and follow-up observations have been performed to further investigate the nature of this γ-ray source. Methods. The Carina region towards the open cluster Westerlund 2 has been re-observed, increasing the total exposure to 45.9 h. The combined dataset includes 33 h of new data and now permits a search for energy-dependent morphology and detailed spectroscopy. Results. A new, hard spectrum VHE γ-ray source, HESS J1026-582, was discovered with a statistical significance of 7σ. It is positionally coincident with the Fermi LAT pulsar PSR J1028-5819. The positional coincidence and radio/γ-ray characteristics of the LAT pulsar favors a scenario where the TeV emission originates from a pulsar wind nebula. The nature of HESS J1023-575 is discussed in light of the deep HESS observations and recent multi-wavelength discoveries, including the Fermi LAT pulsar PSR J1022-5746 and giant molecular clouds in the region. Despite the improved VHE dataset, a clear identification of the object responsible for the VHE emission from HESS J1023-575 is not yet possible, and contribution from the nearby high-energy pulsar and/or the open cluster remains a possibility.