I'll take your first integral, to give you an indication of what types of closed-form expressions you can expect:

$${\cal I}(x)={\cal P}\int_{-1}^{1}\frac{\sqrt{1-t^2}\sin kt}{t-x}dt$$

For $x=0$ this evaluates to

$${\cal I}(0)=\frac{\pi}{2}\left(J_{1}(k)[-2+\pi k H_{0}(k)]+kJ_{0}(k)[2-\pi H_{1}(k)]\right)$$

where $J_n$ is a Bessel function and $H_n$ a Struve function.

For nonzero $x$ the best I (read: `Mathematica`

) can do is a power series expansion around $x=0$, containing all even powers $x^p$. Each term in this series contains the Bessel and Struve functions $J_0(k),J_1(k),H_0(k),H_1(k)$, multiplied by polynomials in $k$ of order $p$. Not particularly useful, but at least higher order Bessel/Struve functions do not appear.

The power series can be written in a compact form in terms of the generalized hypergeometric function ${\cal F}$ $\equiv$ $_{p}F_q$,

$${\cal I}(x)={\cal I}(0)+\sum_{n=1}^{\infty}x^{2n}(-1)^n \frac{\pi k^{2n-1}}{(2n-1)!} {\cal F}\left(-\frac{1}{2};n,n+\frac{1}{2};-\frac{1}{4}k^2\right)$$
The same applies to your second integral, same Bessel and Struve functions in different combinations, similar series of hypergeometric functions.