There are two bible accounts of Gentile mother that came to Christ to ask Him to castoff a demon that plagued her daughter. The first is Matthew:
Matthew 15:21 When Jesus left there, He withdrew to the area of Tyre and Sidon. 22 Just then a Canaanite woman from that region came and kept crying out, "Have mercy on me, Lord, Son of David! My daughter is cruelly tormented by a demon." 23 Yet He did not say a word to her. So His disciples approached Him and urged Him, "Send her away because she cries out after us." 24 He replied, "I was sent only to the lost sheep of the house of Israel." 25 But she came, knelt before Him, and said, "Lord, help me!" 26 He answered, "It isn't right to take the children's bread and throw it to their dogs." 27 "Yes, Lord," she said, "yet even the dogs eat the crumbs that fall from their masters' table!" 28 Then Jesus replied to her, "Woman, your faith is great. Let it be done for you as you want." And from that moment her daughter was cured.
If you look at vs 24 what sticks out to me is the word 'only'. Christ goes on to show us that 'Faith', has the power to jump the divide that lies between the Law and Grace, between Jews and Gentiles.
The next account is in Mark:
Mark 7:24 He got up and departed from there to the region of Tyre and Sidon. He entered a house and did not want anyone to know it, but He could not escape notice. 25 Instead, immediately after hearing about Him, a woman whose little daughter had an unclean spirit came and fell at His feet. 26 Now the woman was Greek, a Syrophoenician by birth, and she kept asking Him to drive the demon out of her daughter. 27 He said to her, "Allow the children to be satisfied first, because it isn't right to take the children's bread and throw it to the dogs." 28 But she replied to Him, "Lord, even the dogs under the table eat the children's crumbs." 29 Then He told her, "Because of this reply, you may go. The demon has gone out of your daughter." 30 When she went back to her home, she found her child lying on the bed, and the demon was gone.
Here the story has a slight change. In vs 27 we see Christ telling the woman that Israel should be 'first' recipients implying that Gentiles would come later. What I see here is that Christ, because of the woman's 'Faith' makes a jump but this time it is a jump in time or priorities.
I like to say that I do not see a conflict in the Scriptures and I would remind all that Mark was writing to the Romans and Matthew was writing to the Jews. There is no real difference in the telling of the account and what occurred and why it occurred.
To the Jews Christ it says, I came here for you but her Faith is stronger then your Law. And to the Romans He said, you too can take part in the promises made to Israel.