Addressing Background Deficiencies In Statement of Purpose: An Initial Solution
If your potential advisor wants to fund you as a PhD student, then it is very likely you will be given admission as their student regardless of whatever background deficiencies you may have. This is especially true now that your potential advisor is aware of those deficiencies. In terms of writing your statement of purpose, it is probably sufficient to say that you worked with Prof. x to create a plan to remedy these deficiencies.