Pretty much anything will run on 4.1v that could run on 4.2, the problem is actually getting it to not charge to 4.2.
The easiest way to charge a lithium battery is with an integrated chip that doesn't have any kind of voltage settings. At the hobby level, it's basically always a TP4506, in commerical projects it's often that, but can sometimes be other chips.
Sometimes more expensive items even use more advanced ones that DO allow voltage settings, and still use 4.2v for some reason!
The easiest way to charge a lithium battery is with an integrated chip that doesn't have any kind of voltage settings. At the hobby level, it's basically always a TP4506, in commerical projects it's often that, but can sometimes be other chips.
Sometimes more expensive items even use more advanced ones that DO allow voltage settings, and still use 4.2v for some reason!