Do recurring evaluations by research funders genuinely enhance research quality and societal impact, or do they merely create unnecessary administrative burdens?

Periodic assessments by research sponsors can improve the quality of research as well as its impact on society, as long as they are well thought out, reasonable, and revolve around metrics that matter rather than mindless reporting. Continuous feedback helps researchers improve their methodologies, align their projects with societal changes, and maintain accountability to their stakeholders. On the contrary, if the requirements become too rigid, too much red tape, or too frequent, they will start to pull researchers away from critical scientific work, fostering an environment that stagnates innovation and progress. In the end, the net positive depends on finding the equilibrium, assessments ought to be relevant, efficient, and devoid of restrictions, adding to the refinement process instead of acting as bureaucratic roadblocks.

It depends! To me, it is even beneficial for researchers to have a self-evaluations of the research. If funders ask for only few periodic evaluations with a resonable amount of required data, not only funder have full control on its objectives and vison but also researcher have a better understanding of its progress. Of course, vigorous recurring evaluations hinder time and effort of both researchers and funders. 

The audits allow fund management to go through their management processes.  They tend to be overbearing administrative demands that can inhibit the conduct of the research and key times and do not enhance the research output.

This depends  on the sponsor. However, the experience with EU grants is that although this may generally enhance research quality or more realistically - grant writing, this will not help in grant approval unless the researcher finds an additional added value to the project proposal (this of course is never asked by the reviewers). This statement is based on personal experience and a concurrent hypothesis.