Turns out there is no evidence that I have been able to find of studies like this in the rural African villages that we at FAVL care about. The closest study I can find is by James Kim, an education specialist now at Harvard. Here's the abstract for one of his papers:
The Effects of a Voluntary Summer Reading Intervention on Reading Achievement: Results from a Randomized Field TrialSee also: Literature Review on the Impact of Summer Reading Clubs
The effects of a voluntary summer reading intervention were assessed in a randomized field trial involving 552 students in 10 schools. In this study, fourth-grade children received 8 books to read during summer vacation, and were encouraged by their teachers to practice oral reading at home with a family member and to use comprehension strategies during independent, silent reading. Reading lessons occurred during the last month of school in June, and 8 books were mailed to students on a biweekly basis during July and August. The estimated treatment effects on a standardized test of reading achievement (Iowa Test of Basic Skills) were largest for students who reported owning fewer books at home, less fluent readers, and minority students. These findings suggest that a voluntary summer reading intervention may represent a scaleable and cost-effective policy for improving reading achievement among lower-performing students.