The Greek word used for mercy is most often eleos(pity, compassion) and for grace is charis(goodwill, lovingkindness, favor).
Mercy and grace are closely related. They may have similar meanings, but they have their own unique characteristics.
As posted previously, mercy is deep and great compassion, release and forgiveness towards a wrongdoer. It is usually spoken of in the context of God not punishing us as our sins deserve. Rather, God full of mercy, in great and deep compassion, releases us from the penalty and consequences from our wrong by pardoning and forgiving us of our wrongs. There has to be a wrong that cannot be rectified and for which adequate compensation has not been made and full satisfaction not paid before mercy can be called into being and given. In its deep and great compassion we find also profound lovingkindness.
Grace shares with mercy the characteristics of unmerited favour. Essential to grace are also the elements of lovingkindness and compassion found in mercy. Grace carries the idea of an unmerited and/or undeserved gifting of a gift or favour to another. The recipient of the grace has not done anything to merit or deserve the gift or favour. It matters not whether a wrong has been committed. Its reach is more extensive and generous than mercy.
As such, it may help to view and understand mercy as a subset of grace, a very special and specific kind of grace, for a specific condition or situation. Mercy proceeds from and is birthed from grace.
The first time grace is mentioned in the Old Testament, it concerned Noah, and the narrative goes like this:
“5 And God saw that the wickedness of man was great in the earth, and that every imagining of the thoughts of his heart was only evil continually.
6 And the Lord repented that He had made man on the earth, and it grieved Him in His heart.
7 And the Lord said, “I will destroy man whom I have created from the face of the earth, both man and beast, and the creeping thing and the fowls of the air, for I repent that I have made them.”
8 But Noah found grace in the eyes of the Lord.”
Here, we find that the word grace encompasses mercy within it.
I believe that the writer of Hebrews had a similar understanding when writing: “Let us then fearlessly and confidently and boldly draw near to the throne of grace (the throne of God’s unmerited favor to us sinners), that we may receive mercy [for our failures] and find grace to help in good time for every need [appropriate help and well-timed help, coming just when we need it].”
Here, again, we have the imagery of mercy and grace being found in and received out of grace.
Paul’s statement and understanding of grace, I believe, is made very clear in Romans 11:6: “And if by grace, then it is no longer by works; otherwise grace would no longer be grace. But if it is by works, then is it no longer by grace; otherwise work would no longer be work.”
Such being the case, if Jesus died at the cross as full payment or full satisfaction for our sins against God, then it would seem to suggest that, in respect of His act:
I continue with my musings.
 See Strong’s G1656 ἔλεος éleos – compassion; kindness or good will towards the miserable and the afflicted, joined with a desire to help them.
See also Strong’s 1653 (ἐλεέω) - have compassion (pity on)
See also Strong’s H7355 רָחַם râcham, raw-kham'- to love, especially to compassionate:—have compassion (on, upon), love, (find, have, obtain, shew) mercy(-iful, on, upon), (have) pity,
And H2617 - חֶסֶד cheçed, kheh'-sed; from H2616 -(loving-) kindness, merciful (kindness) pity.
See Isaiah 54:8 where both words were used.
 See Strong’s G5485 χάρις cháris – good will, loving-kindness, favour
See also Strong’s H2580 חֵן chên khane; from H2603 – kindness, favour, have pity and compassion
 21st Century King James Version
 Strong’s H2580 חֵן chên khane; from H2603
 See also Deuteronomy 34:6–9 especially vs 7 and 9
 Literally: bold freedom See Strong’s G3954 - παρρησία parrēsia – boldness, free and fearless confidence, cheerful courage, boldness, assurance, freedom in speaking, unreservedness in speech
 “take hold of” See Strong’s G2983- λαμβάνω lambánō
 Amplified Bible Classic Edition
 Modern English Translation