Patch all you want but that's going to fix that problum.
The door was installed wrong and will still leak.
Let me make a guess, there's a deck, stoop steps, something that's even or close to it on the other side of that door.
The best way will involve removing that door and most likly some of the siding to totaly water proof the opening and outside wall before the door goes in.
Using an ossilating saw or a Toe Kick saw will give you a much tighter cut then a sawsall.
Post a picture of the outside of that door opening.
Most often the underside of the threshold on the outside is also not done right so it's not fully sealed or supported.
If someone installed any form of a 2X under the threshold it will leak 100% of the time. It sticks out to far and will act like a funnel to direct water under the threshold. Using 1X vinyl lumber works much better.