How to fill holes/missing values in XYZ data / a DEM with QGIS?

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I have equidistant XYZ point (text) data that I used to create a point grid. However, it's not perfect - there are some holes. I then created a DSM raster from it, which obviously has the holes too. The holes range from a single point to larger areas (~50 or more points missing).

How do I patch up those holes? I tried to interpolate the raster, but that reduces the quality, and doesn't fix the zero "holes" - they're just slightly raised afterwards. Ideally I'd like to just fill those zero values with Z values from their neighbouring squares.

Another issue is that the XYZ data set isn't rectangular but like in the third attached image (red path). If I use a fill method, how do I avoid filling up the whole surrounding area?





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