The prayer was not apparently a translation of a pre-existing prayer found in the Sarum Rite - but was a unique combination of several sources, including phrases or concepts from Mark 7:28, the Liturgy of Saint Basil, a Gregorian collect, John 6:56, and the writings of Thomas Aquinas.
[1] In its earliest appearance the prayer followed the confession and absolution and "comfortable words" which were inserted after the Roman Canon of the Mass.
It is a reply from a woman in speaking to Jesus regarding her unworthiness, who said, "but even the dogs under the table eat the children's crumbs".
The version of 1548 and 1549 appear below with modernised spelling: We do not presume to come to this thy table (O merciful Lord) trusting in our own righteousness, but in thy manifold and great mercies: we be not worthy so much as to gather up the crumbs under thy table: but thou art the same Lord, whose property is always to have mercy: grant us therefore, gracious Lord, so to eat the flesh of thy dear Son Jesus Christ, and to drink his blood, [in these holy Mysteries,] that we may continually dwell in him, and he in us, that our sinful bodies may be made clean by his body, and our souls washed through his most precious blood.
But thou art the same Lord, whose property is always to have mercy: Grant us therefore, gracious Lord, so to eat the flesh of thy dear Son Jesus Christ, and to drink his blood, that our sinful bodies may be made clean by his body, and our souls washed through his most precious blood, and that we may evermore dwell in him, and he in us.
A contemporary language rendition is found in the 2019 Prayer Book of the Anglican Church in North America.
But You, O Lord, are unchanging in Your mercy and Your nature is love; grant us, therefore, God of mercy, God of grace, so to eat at this Your table that we may receive in spirit and in truth the body of Your dear Son, Jesus Christ, and the merits of His shed blood, so that we may live and grow in His likeness and, being washed and cleansed through His most precious blood, we may evermore live in Him and He in us.
[3]There is some similarity with the prayer said immediately prior to communion in the Roman Rite Mass: Domine, non sum dignus, ut intres sub tectum meum: sed tantum dic verbo, et sanabitur anima mea which is translated: "“Lord, I am not worthy that you should enter under my roof, but only say the word and my soul shall be healed.
Either the Prayer of Humble Access or the Domine, non sum dignus is recited prior to the Communion of the Faithful.