The following assumes it is leaking where the horizontal part of the white flush pipe meets the back of the bowl.
1. The pipe is sealed into the spigot on the back of the bowl by a flush pipe cone. These can become hard with time and fail to seal properly.
2. They come in two types, plastic and rubber. Both types have fins around the outside of the body, and it is these fins flexing and pushing against the inside of the spigot which forms the seal.
3. In my opinion, the plastic ones are useless because there is not enough flex in the fins. The rubber ones are fine. This link shows what I mean
Internal White Rubber Flush Pipe Cone - 08000681 - https://www.plumbers-mate-sales.co.uk/internal-white-rubber-flush-pipe-cone---08000681-2193-p.asp
4. Its unlikely you need all the pipework in the Screwfix example you have shown, and in any case fitting it would require disturbing the joint at the cistern.
5. I would suggest you get the flush cone noted above or something similar. Then:
5a. Turn off the water supply to the cistern. Not essential if its a syphon operated one, which it probably is.
5b. Flush the WC.
5c. Pull the white pipe out of the back of the bowl. There is likely to be a small amount of water, so old towels.
5d. Thoroughly clean and descale the inside of the spigot and the outside of the pipe.
5e. Slip the new flush cone over the pipe, with the narrow end of the cone level with the open end of the pipe. Mark on the pipe where the other end of the flush cone comes to, another mark 12 to 15 mm behind the first, then push the cone so that its back (wider end) coincides with the second mark.
5f. Push the cone and pipe together into the spigot. Make sure the back of the cone is somewhere between the two marks.
5g. Turn the water back on, allow the cistern to refill and test.
5h. If it still weeps, remove again, put Plumber's Mait into the first two fins and try again.
5i. If still no good, remove again, smear some clear silicone over the pipe and try again, but wait 24 hours before flushing.
6. You might find that removing the old cone, and cleaning it as well as the spigot and pipe, then refitting with Plumber's Mait and possibly silicone as described above will fix it.
