Can hyperconjugation take place between lone pairs of an atom attached to an sp2 carbon?
Hyperconjuation is the interaction of electrons in sigma orbitals (in $C-H$ or $C-C$) with an adjacent empty p orbital or $\pi$ orbital. What you are describing is not hyperconjugation, but resonance; more specifically, the chlorine shows the $+R$ effect.

Now if you draw the resonance structures of this molecule you will realize that the $C_3$ has a negative charge on it in one canonical form. This means, in the actual molecule, that $C_3$ carbon has a partial negative charge, which is why the hydrogen from $HBr$ preferentially bonds to that carbon.

A more acceptable explanation would be that, when the hydrogen bonds to $C_3$, the resultant intermediate carbocation (positive charge on $C_2$) is stabilized by the $+R$ effect of chlorine, resulting in a racemic mixture of $2-bromo-2-chlorobutane$ to form as the major product.