Here's a conference-lecture by Dan Zahavi (of the "Center for Subjectivity Research" at the University of Copenhagen/Danish National Research Foundation) that asks whether it's a good idea to try to "naturalize" phenomenology.
He distinguishes early on what Flanagan means by phenomenology (referring to Owen by name), i.e. reports on what things seem like to us, and what Husserl, Merleau-Ponty, and the rest of that tradition meant by it.
Husserl's founding motivation was against psychologism, which involved understanding phenomenology as introspection by the kind of being already roughly understood by this science of psychology. Following Frege, he argued that you can't understand, for instance, a number by analyzing it as an occurrence in the head; this approach completely prevents understanding of the system of logic involved in mathematics. Likewise, for Sartre, phenomenology is not introspection, which implies looking within oneself, because for Sartre, the self is not the object of the phenomenological gaze except during reflective consciousness, i.e. when analyzing our attitudes towards the self. We are instead describing the world, and we can't prejudge that things as experienced will line up with categories predefined by science.
Flanagan's approach in our discussion with him seemed pretty reasonable: when phenomenology makes claims about things that empirical science also has something to say about, you need to compare those, and the science is typically going to win out. I am in a sense free, in that for the purposes of planning my own actions, it works best for me to act this way, but when designing a system of punishment for society, or reproaching myself retrospectively for my failings, or predicting even my own future behavior (e.g. I set up some rewards for myself should I achieve my goals) or any number of other purposes, the science that acknowledges determinism should win out.
Still, the way Flanagan put it still seems troublesome if you're really committed to some form of pragmatism or transcendental phenomenology: you have to decide whether the scientific or the phenomenological claim is true. For William James, truth itself is defined by experience, such that empirical truth is something that we understand through familiar patterns of empirical verification (i.e. we go look at something, or at least we understand that one could go look at something), while the truth of other types of statements (such as the one about our personal freedom, or religious claims) is established in other ways, and so, in effect, "truth" means something somewhat different when applied to those claims. So it may not really be sensible to ask which of two very different types of claims is true, assuming them to be mutually exclusive. If truth is defined as a matter of some kind of verification process (not defining "verification" narrowly here as in the logical positivists), then two truth claims involving very different kinds of verification aren't necessarily comparable at all; they're in effect talking about different things. You can say, as Flanagan did, that the two facts about free will are appearance (our phenomenological experience of our freedom) and reality (what science tells us), the grounds for this epistemic hierarchy are within the assumption of science, i.e. of naturalism, whereas a phenomenologist like Merleau-Ponty is going to say that naturalism itself has only phenomenology to justify it, so the phenomenology can't just be dismissed as "mere appearance" even when it seems to conflict with some verdict of science.
All of this may be preliminary to Zahavi's first step. He's concerned with various attempts people have made more recently to make phenomenology serve empirical science. I'll let you watch the video to see his presentation of the issue, though.
(Thanks to DMF for referring me to this.)