Only happens with any pressure on the pedals.
In normal use it clinks twice per pedaling cycle.
More specifically, it does it any time there is any pressure on the pedals and I suddenly release that pressure.
Tried adjusting these 4 screws without success. Anything gives ?
In response to your suggestions:
The crank itself does not appear to be loose.
The clinking is present on other sprockets / chainrings
Standing up out of the saddle makes no difference
What does make a difference is back-pedaling with equal force on both pedals - the clinking almost disappears. If unequal forces when backing off on the pedals then it behaves as described above in the OP text.
More to the above, I have turned the bicycle upside down and hand pedaled backwards. I can feel and hear multiple nano-clinks. This together with the previous point makes me think it has something to do with the bottom bracket.
I shall take off the pedals too and clean / grease them; one of them won't freely move, the other does, and makes quite a sound - as off worn out bearings.
Oh, and I am unable to exert enough pressure by hand in order to reproduce the issue with the bike upside down like that.
The teeth on chainrings look fine to me.
Extra
FWIW: this bike has never done very well. I had to take it to the shop when it was brand new some 7-8 years ago for some front chain related adjustments.
This clinking has developed some months ago along with the chain going off the chainring especially on the smaller chainrings. I have since cleaned the back parts (loooots of grime there) and the situation has improved a lot. Chain won't slip anymore on largest chainring, it still does it on the smaller ones. Not sure if this is related to the clinking though. My bicycle is optimized to run on largest chainring and a few rear sprockets. Anything else it makes extra noises, at the very least.