I recently clipped a raster layer to a smaller extent. When I import this layer and assign it the same CRS as other layers (EPSG:4326, WGS 84) it does not project with them. The original pre-clipped raster layer projects correctly as do CSV data points that I have imported. I checked the Project properties and it also has EPSG:4326, WGS 84 assigned. Is it possible that the clipped layer is in meters instead of degrees? How would I determine if each layer is using either meters or degrees?
You may want to try using the Raster>Clipper and a mask polygon in order to clip your raster while maintaining its spatial validity.
All you need is your base raster that's going to be clipped and a vector polygon of the area you want to clip from your raster. Then you can go to Raster>Extraction>Clipper.
In the Clipper, set the name and location for your raster clip output, select Mask Layer as your Clipping mode, set your clipping polygon as the Mask layer, choose whether or not to use the extent of the mask layer as the extent of the clip output (you probably want to), and choose whether or to keep the input resolution or set it to set it yourself (you probably want to keep the input resolution).
You should end up with a clipping of your original raster that maintains its spatial reference.
Alternatively, if you don't have a polygon layer to use and would rather just draw a quick polygon to crop your raster with:
Instead of using a Mask Layer, you can use the Extent option in the Clipper tool and just draw a box or type in the bounding coordinates of what you'd like to clip out.
With the Extent option toggled on, you can click+drag a box to use for clipping. Once you get some coordinates showing up in the x,y fields of the tool, you're all set to clip.