Do external electronic shells influence internal ones?
Not only do "outer" electrons impart some shielding to "inner" ones, but an analytical technique is based on that fact.

In X-ray photoelectron spectroscopy, binding energies of electrons just below the outer shell are measured.  These binding energies are typically hundreds of electron volts, roughly 280-300 eV for carbon as a typical example.  An important feature is that this binding energy is measurably not fixed for a given element, rather is varies according to how the atom is chemically bound.  As an example, the 1s electrons in carbon lie at about 285 eV binding energy in hydrocarbons, but shifted by as much as 10 eV by bonding to fluorine (see here), which is more electronegative and draws electrons away from the carbon 2s and 2p  shells.  Similarly, the 1s electrons in oxygen range from about 529 to 534 eV in various oxides, with the higher values tending to occur in nonmetal oxides (see here).