Re-insulating basement and concerned about correctly addressing moisture currently on sill plate that is resting on foundation wall

Question

I'm attempting to finish a basement that currently just has blanket wrapped insulation.

I pulled off the blanket wrapped insulation enough to inspect it and found that there was a lot of moisture. The sill plate on top of the foundation was particularly wet and black mold is growing.

It looks like the house wrap comes in from the exterior over top of the foundation wall; under the sill plate which is sitting on what looks to be "ProPINK ComfortSeal Sill Gasket". The house wrap is tuck taped to more house wrap which continues down the interior side of the foundation wall into the floor slab, behind an existing membrane coming out from the floor.

The house wrap has what looks to be deliberate sections cut out of it so there are 3 squarish areas removed per wall. I'm a little confused as to why this was done? The house wrap around and particularly above the cut out sections has a lot of condensation on it. There is frost and ice forming on the exposed concrete. The sill plate on the top of the foundation is getting wet, and in some areas, wet right through to the top side of the plate and looks to be wetting the osb its touching. The house wrap under the gasket is soaked in areas and allowing black mold to start growing on the plate above it. The house wrap doesn't look to be wet on the side facing the concrete, but more on the interior facing side.

I'm going to add 2 inches of ridgid foam directly to the concrete, then frame my wall right against that and add pink insulation (R-12 or R-14). I would bring the foam right up to the floor joints and spray insulating foam sealant, between the sill plate and the foam to seal that off. Then also fill the osb joist cavities with the same 2" foam and foam sealant. (I will remove all the house wrap from the foundation wall, leaving just a bit coming out from under the sill plate)

However I want to make sure I understand the current moisture problem and know that the new work will address it.

My inexperienced theory is that the holes in the house wrap allowed condensation to form and that moisture migrated up to affect the sill plate and osb? Would condensation form directly on the interior side of that house wrap if the holes where not present? Is there still a danger of condensation directly under the sill plate where its sitting on the house wrap?

Will the rigid foam application likely address this issue or is there something else I should be considering?

Images added to show better what words cannot always describe clearly.

Updating with additional photos.

I marked (with x's) on a little diagram where particularly wet spots on the sill where)

I've only pulled down half of the wall wrap and have pictures of what the existing wrap looks like in the next room.

Answer 1

After reading your post several times, here's what I think is going on.

Your concrete basement wall is cold, with the cold coming from the outside. This is the only thing that makes sense and is consistent with your comment that "There is frost and ice forming on the exposed concrete". The moisture is coming from the interior of the basement, hitting the cold concrete and then condensing into water or frost, depending on the temperature of the concrete.

The placement of the house wrap (which is not a vapor barrier) on the inside of the concrete wall does not affect this. The moisture is condensing out of the air on the cold concrete or the cold house wrap, whichever it encounters first.

So you need to find a way to 1) keep the concrete wall warm (probably not practical, as this would entail adding foam insulating boards to the outside of the concrete wall), or 2) keep the moisture away from the wall.

This https://buildingscience.com/documents/digests/bsd-106-understanding-vapor-barriers is the best reference I've come across for how to handle different wall construction and insulation situations.