Exaudio is one of my favorite words in Latin because it doesn’t translate too well. Audi means to hear or listen. Exaudi means…well, in the litany of the saints it’s usually translated as “graciously hear” so that it doesn’t simply repeat the audi immediately before (Christ hear us…Christ graciously hear us).
To get the sense, compare with specto and exspecto. Specto is simply to look, while exspecto means to be on the lookout, to scan the horizon like a sentry for the anticipated arrival of a thing–hence, mundanely, to await. Exaudio is the same thing but with listening, like when the fainting damsel in the horror movie is hiding in the dark house listening for sounds of the intruder. Colloquially we might say “Listen up, what I say next is important!” or something like that. That’s the kind of attentive listening we are asking of God, not just that He happen to hear our prayers.
I probably said all this in a Psalms post last year (1-25), but it bears repeating!
“Ad te Domine clamabo” (Psalm 27)
 Psalmus ipsi David. Ad te, Domine, clamabo; Deus meus, ne sileas a me : nequando taceas a me, et assimilabor descendentibus in lacum.
A Psalm of David himself. To You, O Lord, shall I cry; my God, lest you be silent from me never keep silent from me, and I shall be like those descending into the pit.
 Exaudi, Domine, vocem deprecationis meae, dum oro ad te, dum extollo manus meas ad templum sanctum tuum.
Hear, O Lord, the voice of my deprecation when I pray to You, when I extol my hands unto Your holy temple.
 Ne simul trahas me cum peccatoribus, et cum operantibus iniquitatem ne perdas me; qui loquuntur pacem cum proximo suo, mala autem in cordibus eorum.
Lest then You drag me off with sinners, and with them working injustice You destroy me; who speak peace with their neighbor but evils in their hearts.
 Da illis secundum opera eorum, et secundum nequitiam adinventionum ipsorum. Secundum opera manuum eorum tribue illis, redde retributionem eorum ipsis.
Give to them according to their works, and according to the wickedness of their inventions. According to the works of their hands bestow to them, return their retribution to them.
 Quoniam non intellexerunt opera Domini et in opera manuum ejus; destrues illos, et non aedificabis eos.
For they have not understood the works of the Lord and in the works of His hands; You will destroy them and not rebuild them.
 Benedictus Dominus, quoniam exaudivit vocem deprecationis meae.
Blessed Lord, for He has heard the voice of my deprecation.
 Dominus adjutor meus et protector meus; in ipso speravit cor meum, et adjutus sum : et refloruit caro mea, et ex voluntate mea confitebor ei.
The Lord my helper and my protector; in Him has hoped my heart and I have been aided; and flourishes again my flesh and out of my will shall I confess Him.
 Dominus fortitudo plebis suae, et protector salvationum christi sui est.
The Lord, the strength of His people and the protector of the salvations of His Christ is He.
 Salvum fac populum tuum, Domine, et benedic haereditati tuae; et rege eos, et extolle illos usque in aeternum.
Make safe Your people, O Lord, and bless Your heritage; and rule them and extol them even unto the eternal.
Forgot to add: don’t ask me why it’s ipsi in the superscription. It’s either some random audience, a mistake where ipsius is called for, or some kind of code/formality.