The Prophet ﷺ said, "Whatever dog or falcon you have trained, then released while mentioning the name of Allah, eat from what it catches for you."I asked, "Even if it kills the prey?"He replied, "If it kills it but does not eat any of it, then it has only caught it for you."
حدثنا عثمان بن أبي شيبة حدثنا عبد الله بن نمير حدثنا مجالد عن الشعبي
أن النبي ﷺ قال ما علمت من كلب أو باز ثم أرسلته وذكرت اسم الله فكل مما أمسك عليك قلت وإن قتل قال إذا قتله ولم يأكل منه شيئا فإنما أمسكه عليك
[Its chain is daif] Reported by Tirmidhi, Book of Hunting, Chapter on What Has Been Said Regarding Hunting with Hawks, no. 1467, from the hadith of Mujalid, and he said, "We know it only through the hadith of Mujalid, and Mujalid is daif." His hadith has corroborating mawquf reports found in Bayhaqi, 9/235, 238.