It has been shown that high quality fine-tuning boosts the performance of language models, even if the size of the fine-tuning is small. In this work we show how highly targeted fine-tuning improves the task of hate speech counter-narrative generation in user-generated text, even for very small sizes of training (1722 counter-narratives for English and 355 for Spanish). Providing a small subset of examples focusing on single argumentative strategies, together with the argumentative analysis relevant to that strategy, yields counter-narratives that are as satisfactory as providing the whole set of counter-narratives. We also show that a good base model is required for the fine-tuning to have a positive impact. Indeed, for Spanish, the counter-narratives obtained without fine-tuning are mostly unacceptable, and, while fine-tuning improves their overall quality, the performance still remains quite unsatisfactory.