Ada-1 doesn't give you the handcannon, it drops when you complete the forge. The frame disappears at this time.
I can see you have 'Kindled Orchid' in your collections, this is the weapon that comes from the Legendary Hand Cannon Frame, so you must have received it at least once. Is the weapon not in your inventory, or in the postmaster?
After you complete the forging, you can rebuy the frame from Ada by using Modulus Reports instead of the rarer Ballistic Logs. Yes, once you rebuy, you need to start from the precision shot step. Is it definitely showing as Abandoned?
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I received the orchard from the dark matter and not the forge so maybe because I already had it it did not show I just completed the legendary machinegun, so is the hammerhead the machinegun you get after you ignite the forge?