Trying to get rid of an airlock

[Dark Eyes]

Every year or so, I seem to need to clear the gunk out of my kitchen sink waste pipes, which also take the waste feeds from the washing machine and dishwasher (see attached photos), which I do, after advice from a plumber, by removing the trap under the sink and sucking out the water and gunk with a Vax (whilst blocking off the washing machine waste pipe behind the cupboard).
It works, and when everything is connected back up, the sink drains at an acceptable rate, and the washing machine doesn't cause the waste pipe to overflow when draining.
However, this year, I have done exactly the same thing, but I now have an airlock somewhere past the trap, as the sink drains really slowly, and it burps for a while after it has finally cleared. I have removed and reattached the trap 3 times to see if I can clear it, but it stubbornly remains. If I release the connector on the horizontal pipe leading to the back of the cupboard, air hisses, water drips, and the sink drains quickly; but as soon as I tighten it back up, the air lock returns.
Does anybody have a suggestion as to how I might clear it permanently?

 

[Newboy]

You have a partial blockage further downstream

 

[Dark Eyes]

This morning, I decided to 'Vax' it again via the horizontal pipe after the trap, and there was a thumb sized lump of clotted washing powder in the grey water. After connecting it all back up, the sink was draining freely with no siphoning or prolonged burping, so I ran the washing machine and as soon as it started draining the vertical waste pipe started overflowing...

I then removed the back of the cupboard, and undid the trap to see if it was clogged, but it was really clean, so I decided to 'snake and Vax' the waste pipe from that entry point, and although there were no noticeable lumps in the grey water, the washing machine is now draining freely as well.

How long it will last, I don't know, we switched to washing powder about a year ago, as I was told that the liquids were causing the pipe to gunk up and get blocked, but the powder seems to be far worse, as I have had to 'Vax' the pipe about 3 times in the past 6 months.

 

[Madrab]

The usual trouble with washing machines is that people use too much detergent/powder. It then doesn't get diluted properly and can then re-constitute around a restriction or a waste run that has too many elbows and/or not enough fall.

Bad waste run design is always a problem and why it's always better to use swept bends rather than tight elbows in a sink/appliance waste run and minimise the number or turns.