$k$ wave number expectation value

It is important not to confuse operators, states, and wavefunctions.

Let us have a state $|psirangle$ with position-basis wavefunction $langle x|psirangle= psi(x)$. Inserting a complete set of momentum eigenstates $|krangle$, or equivalently expressing the identity operator as
$$ {mathbb I}=int frac{dk}{2pi} |krangle langle k |, $$ we can write $$ langle x|psirangle = int frac{dk}{2pi} langle x|krangle langle k |psirangle= int frac{dk}{2pi} e^{ikx} phi(k) $$ where $phi(k)= langle k|psirangle$ is the wavefunction for the state $|psirangle$ in the momentum basis, and $langle x|krangle = e^{ikx}$ is the wavefunction for the state $|krangle$ in the $x$ basis.

Now recall that in the $x$ basis the momentum opertor $hat p$ acts as $$ langle x|hat p|psirangle = -ipartial_x langle x|psirangle $$ but in the momentum basis we have the much simpler $$ langle k |hat p|psirangle= k langle k|psirangle. $$ So again using $$ {mathbb I}=int frac{dk}{2pi} |krangle langle k | $$ we can write $$ langle psi|hat p|psirangle = int frac{dk}{2pi}langle psi|hat p|krangle langle k|psirangle =int frac{dk}{2pi} klangle psi|krangle langle k|psirangle= int frac{dk}{2pi} k phi^*(k)phi(k). $$ There is no hat on the $k$ in this last expression because it is a number not an operator.