So fluorine does have the ability to hydrogen bond. However, they are usually transient and depend significantly on the species that is donating the H-bond along with some other factors (i.e. what phase the molecules are in, temperature, etc...)
Fluorine is so electronegative that it holds its electrons much "tighter" then other atoms, making it a poor H-bond acceptor. Oxygen, being less electronegative, is a better H-bond acceptor since the electrons aren't held onto as tightly, allowing them to interact more with other atoms.
Since bond energy is "the average value of the gas-phase bond-dissociation energy for all bonds of the same type within the same chemical species. The larger the average bond energy, per electron-pair bond, of a molecule, the more stable and lower-energy the molecule" (IUPAC WEBSITE), I would suggest that the H-bond with oxygen has a higher bond energy, as it is the stronger interaction of the two.
The OP requested some sources, in which I failed to do initially, so I apologize for that but here they are.
J. Am. Chem. Soc. 1983, 105, 10, 3206–3214
Tetrahedron, Vol. 52, No. 38, pp. 12613-12622, 1996
J. Am. Chem. Soc. 1962, 84, 16, 3221–3222
Here is a quote from reference 2.
Theoretical calculations variously estimate 5 the strength of a F...H bond to be between 2 to 3.2 kcal mo1-1.
This can be compared 6 to an O...H hydrogen bond which is typically between 5 - 10 kcal mo1-1. Consistent
with this the electrostatic influence of fluorine is approximately half that of oxygen 7. Thus the greater
electronegativity and lower polarisability of fluorine over oxygen, suppresses its electrostatic influence and
renders it a poorer hydrogen bond acceptor.
The last reference discusses hydrogen bonding in halides, not too sure how much it compares them to oxygen bonding but it is certainty worth taking a quick look.