Patching gap in drainage channel
Plain old mortar (or concrete with only small aggregate) would be my choice - it does not need to be a perfect seal. If it's dry when you are trying to do the work, pre-wet the area you are patching for a better bond. You'll also want to scrub off as much of that decaying organic material as possible. Use a fairly stiff mix and beat it into place.
It needs to be wetted out, not dry, but you want a mix you can place, not one you pour. Slapping or beating it once placed (equivalent to vibration but on a reasonable DIY budget) will help get the mix in good contact with the old concrete without having to have a very liquid mix. Apart from being hard to place, a too wet mix is weaker.