Pedrell's edition is here.
Yes, you can call this the 4th tone. Note the points of imitation on B and E; notice that the dux and comes of the opening point emphasise C & F respectively before falling back to the final. Subsequent points (such as at the start of page 2) have similar incipits.
Note also that IV (A major) does not appear in cadences, only in the middle of phrases, and it normally leads to ♭vii. (That's to say that, in the context of this piece, A major is not stable.) Cadences using A tend to take the form of I-iv (occasionally iii°-iv, as in ms 13-14) and iv-I (notably the final cadence, but also at the start of the third system on the first page of Pedrell's edition, for instance). Cadential A is always minor.
The third system, immediately after your example, is where the sense of A as tonic starts to be seriously undercut. In addition to the plagal cadence at the start of the system, notice that the middle of the system prolongs A minor in a purely diatonic setting, and notice the cadential approach to iv is via a 6-5 appoggiatura (i.e., ♭iii6-♭II6-iv). This is a bit of an exceptional situation, but it does not make for a stable landing at all.
The third and fourth bars of the fourth system introduce a cadence that is going to show up at critical points during the piece: ♭vii6-I. iv follows I at the start of the subject entry, but in 2nd inversion, i.e., not at all stable and leading back to I. You can see here the shape of the Phrygian cadence as it came to be used in functional harmony, that is, as iv6-V-i, but the placement of the cadence relative to the phrasing and the undercutting of the mode's iv (A minor) by inversion leave the Phrygian cadence in its modal form here.
The Phrygian cadence shows up again, for example, in the middle of the third and fourth systems and the start of the fifth system on the second page of Pedrell's edition, and it doesn't always move to A (the instance in the fourth system moves to an augmented chord on F, for example).
So to summarise:
- Points of imitation start on B and E, and tend to emphasise C and F as descending leading tones.
- Important cadences take the form of I-iv, iv-I and ♭vii6-I.
- IV is used only in passing, generally en route to ♭vii.
- 5 as a root is unstable and generally avoided, hence 4 as the tenor, but, as important as 4 is, it is usually undercut by not being given a tierce de Picardie in cadential contexts.
- 1, when used as a root, is generally given a tierce de Picardie, and generally functions either as V/iv or I. It is further reinforced as a final by having two types of cadence: a mode-specific "perfect" cadence (♭vii6-I) and a plagal cadence (iv-I).
Cabezón's career overlapped the careers of the Elizabethan composers, and he died during Monteverdi's childhood, so his work is right on the cusp of the Baroque, and some of the strains that led to the Church tones being relegated to the periphery are starting to show: the sense of modal polyphony that saw octave-and-fifth chords as perfect and used them to define cadences was already in the past; the use of functional relations in a triad-based vocabulary was starting to peek through. Cabezón establishes his mode by reliance on constraints and formulae, both cadential and melodic (including how musica ficta is used, hence the avoidance of C♯ at cadences).
Modal music never entirely went away. You can see Hypophrygian mode incorporating a wider tonal range, but using very similar conventions and constraints to establish the mode, in Bach's 6-voice prelude on Aus tiefer Not schrei' ich zu dir from the Clavier-Übung.