MCP4921 audio: removing output DC Bias
You need to change your software so that between outputs of the audio you set the DAC to the mean voltage of the audio - presumably half-scale (0x80).

Then the audio will not be on a pedestal.

Most of these types of DACs use unsigned binary coding.  To use it for audio you need to offer the audio by half-full scale so that you can go either positive or negative relative to the zero level.

Between audio clips you need to set your DAC to hat same level.

It doesn't look like you need any hardware changes.