Shimano STI does not shift up/to lower cogs Typ: Shimano RS505 2x11 hydraulic

Problem: Up shifting is possible to the 5th or 6th cog, after this the STI does not give the click feedback and the derailleur does not move to the next smaller cog.
If the cable is under more tension the STI shifts normaly and makes a proper click on every shift.
The derailleur is cleaned and has full tension in all positions. I tried a different derailleur and got the same result.

When did the problem occur: After a muddy tour the shifting got worse and inaccurate. Today I cleaned the bike with pressure water. I was always far enough away from the STIs, but maybe some water got into it. After that the derailleur started to stuck.

What did I try so far?: I pulled the rubber hoods down and put some grease into the STI, however it did not get better yet.

Any chance to repair the STI?
Sounds like the shift cable have become contaminated by grit or mud and are hanging up or there is excess friction, which stops the derailleur pulling the shifter onto the next highest gear position.

When you give the cable a little extra tension it overcomes the friction allowing the shifter to shift properly.

You can try removing the shift cable from the housing, cleaning and lubing it. If that does not fix the problem you can replace the housing. Start with the section at the derailleur end as it is closest to the ground an has the highest chance of getting something in it.