On the question “Is there any research showing a correlation between wealth and academic performance?”, interestingly in America, we have an extensive parochial school educational system. Here is a report with statistics as available here, to quote:
Math and reading are included in the most recent Digest of Education Statistics, published in February 2021 by the National Center for Education Statistics. The NAEP science scores were released Tuesday in conjunction with a press conference.
“Every year for the last 20 years, Catholic schools have outperformed public schools on NAEP tests—reading, math, science, computer literacy, geography, history,” said Sister Dale McDonald, PBVM, director of Public Policy and Educational Research for the National Catholic Education Association. “We’re happy to have our achievement validated by an outside, public, federal agency.” …NAEP, also known as The Nation’s Report Card, assesses academic achievement using the frameworks developed by the National Assessment Governing Board. The assessment measures student progress in grades 4, 8, and 12.
Now, Catholic schools are part of the private school population but have fees generally accessible by a large portion of middle class Americans, who would not consider themselves as wealthy.
As such, while there may be an overall seemingly significant correlation between wealth and academic performance, clearly wealth is not a causative variable, given the statistics presented. More likely, it is just a proxy variable for other beneficial characteristics including hard work, discipline and family support, also demonstrated in the more successful wealthy families, that generally promotes good academic performance.