Radiotherapy with very high energy electrons has been investigated for a couple of decades as an effective approach to improve dose distribution compared to conventional photon-based radiotherapy, with the recent intriguing potential of high dose-rate irradiation. Its practical application to treatment has been hindered by the lack of hospital-scale accelerators. High-gradient laser-plasma accelerators (LPA) have been proposed as a possible platform, but no experiments so far have explored the feasibility of a clinical use of this concept. We show the results of an experimental study aimed at assessing dose deposition for deep seated tumours using advanced irradiation schemes with an existing LPA source. Measurements show control of localized dose deposition and modulation, suitable to target a volume at depths in the range from 5 to 10 cm with mm resolution. The dose delivered to the target was up to 1.6 Gy, delivered with few hundreds of shots, limited by secondary components of the LPA accelerator. Measurements suggest that therapeutic doses within localized volumes can already be obtained with existing LPA technology, calling for dedicated pre-clinical studies.