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I am fairly new to this and have a question as I always get distortions as soon as I try to extrude any verts of this cylindrical shaped part of the mesh or when I try to connect Verts of this part with other verts from parts of the rest of the model.

The Subdivision Surface Modifyer is causing the distortions. As soon as I switch it off, it's gone. I applied All, flipped the normals (and flipped them back afterwards as it had no effect), Merged by Distance, tried adding a crease around the edge loop, added additional edge loops etc.

Has anyone any idea what is causing these troubles?

enter image description here

I added all the faces around the Edge Loop at the back and the distortions on the outside of the cylinder are finally gone indeed, but there is a white ring clearly visible now inside the cylinder.

enter image description here Is there any chance to get rid of this? Not quite sure, what is causing this.

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It's the expected behaviour: by extruding that vertex you're creating a "non-manifold" geometry. Check this answer for understanding what is manifold and why you should avoid non-manifold geometry.

What is non-manifold geometry?

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