The specific program that is being cut is the 21st Century Community Learning Centers initiative.
Mulvaney's claim is a bit misleading, as providing food is not the primary purpose of CCLCs. The NYT fact-checked the claim and says (Links to their sources not included):
While Mr. Mulvaney specified after-school meal programs, it’s unclear what he is referring to because the government program that does fund after-school nutrition is not specifically at risk (nor does it purport to “help kids do better at school”).
The budget does propose to cut 21st Century Community Learning Centers, which fund after-school programs for about 1.6 million children in high-poverty areas. While the vast majority provide snacks and meals, feeding children is not their core mission.
“It’s something they do, but let’s not miss their primary purpose,” said Heather Weiss of the Global Family Research Program, who has evaluated after-social programs for decades. “They were set up to provide safe environments for kids with learning opportunities of all sorts.”
Looking at the primary purpose, Mr. Mulvaney is wrong that there is no evidence of outcomes. While early research found no evidence of educational outcomes, Dr. Weiss and others have disputed their methodology and say they were conducted before the programs were up and running.
More recent valuations show these programs do improve student performance. According to the community learning centers’ latest publicly available performance report, from 2011, the program hasn’t always met its annual performance targets, but participants did improve math and English grades by 30 to 40 percent every year across all grade levels.
Following is an overview of some studies.
The Afterschool Allience has an overview over a large number of studies, including a short overview of the result. They summarize:
The data and conclusions from these studies suggest that quality afterschool programs have a positive impact on a number of measures of student
academic achievement, positively affecting behavior and discipline and helping relieve parents’ worries about their children’s safety.
There has been an evaluation of the program in Louisiana in 2003 showing positive results:
Participants (30 days or more) showed a significant improvement over nonparticipants on core ITBS scores, with an impact score of 2.2l NCEs (p < .01). Those who attended 60 or more days and 90 or more days experienced nearly the same academic impact over nonparticipants on core scores (2.44 and 2.68 NCEs respectively, p < .01 for each). A Profile of the Evaluation of 21st Century Community Learning Centers—Louisiana
There has also been an evaluation in Chinatown YMCA in 2003 showing positive results:
The treatment group demonstrated significantly higher scores (p < .05) than the comparison group on improvement in turning in homework on time, improvement in satisfactory homework completion, improvement in academic performance, and improvement in attendance.A Profile of the Research Study of 21st Century Community Learning Centers—Chinatown YMCA
A national study in 2005 showed no benefit:
This study finds that elementary students who were randomly assigned to attend the 21st Century Community Learning Centers after-school program were more likely to feel safe after school, no more likely to have higher academic achievement, no less likely to be in self-care, more likely to engage in some negative behaviors, and experience mixed effects on developmental outcomes relative to students who were not randomly assigned to attend the centers.When Schools Stay Open Late: The National Evaluation of the 21st Century Community Learning Centers Program -- Final Report
A national performance evaluation in 2011 showed that while the goal that 48% of participants increasing math grades was not met, 38% showed increased grades in 2011; 40% increased grades in english (goal: 48%), and 74% reported improvements in homework completion (goal:83%).
An evaluation in California in 2012 found few benefits:
Although some minor positive and negative findings were found, the overall effects of the ASES and 21st CCLC programs were neutral.Independent Statewide Evaluations of the After School Education and Safety Program and 21st Century Community Learning Centers Program
There were however positive findings for subgroups:
African American, special education, and “far below basic” students who attended their after school program frequently were found to perform better on academic measures than students who did not participate.
AIR summarizes 2 year evaluations from Texas, Washington, Rhode Island, and New Jersey:
Participating regularly in the 21st CCLC program was consistently associated with higher state assessment scores in reading and mathematics. The effects were small but consistent with expectations for an intervention lasting, on average, 10 hours per week
Little et al of the Harvard Family Research Project have looked into after-school programs in general and conclude that they improve academic achievement.