The conceptual relationship between mercy and pity is a genuine philosophical issue and it deserves investigation. Following your pointer, I shall not bring euthanasia into the discussion.
On the surface, mercy is a matter of action ('an act of mercy') while pity is an emotion. Mercy can be motivated by pity but they belong to distinct categories in the philosophy of mind and action. Let's delve a little deeper.
Mercy and supererogation
Excuse the word, 'supererogation'. I use it because it often comes into play in discussing mercy. The idea is that mercy is undeserved in the sense that if I have done something wrong without valid excuse, I am not strictly entitled to mercy. I cannot claim it as a right or an obligation. Rather, mercy is bestowed gratuitousy on me, independently of what I deserve. For instance, I might steal from a friend. My friend would be justified in reporting me to the police and bringing me to court. He or she would be morally and legally entitled to apply justice in this way. Instead, he or she forgives me, allows me to return the item and closes the matter, and in doing so shows mercy. It is not necessary for pity to enter the picture. A broad variety of motivations may explain an act of mercy.
Mercy and punitive leniency
In this case, mercy is a matter of imposing less than the deserved punishment on an offender but against a certain background. In forgiving me in the example above, my friend might have no concern to alleviate my condition or to contribute to my well-being. In this second kind of mercy, in contrast, such a concern must be present and derive from pity or compasssion for the offender.
Pity has just put in an appearance. Note that if mercy is motivated by pity, they must be conceptually distinct in some way. Mercy cannot be identical with what motivates it.
Pity covers, I think, two emotional states: pain at undeserved bad fortune and pain at deserved bad fortune. I can feel pity for someone who is (say) in prison for a crime for which they deserve (in my opinion) to be punished. While I think they have got their just deserts, I can still as a fellow human being feel pain at the deprivations they are suffering - tight restriction of movement in their cell, little access to the outside world, the roughness and bullying of prison life, etc. This would appear to me an appropriate moral emotion in the circumstances. I mean appropriate in view of our shared understanding of the nature of morality, and not as mere personal idiosyncracy.
Equally I can feel pity for someone who has done nothing wrong and in that sense does not deserve their bad fortune. I can feel pity for the young woman who develops ovarian cancer or the pensioner who has been scammed out of their life savings. In the circumstances, just as above, and free from mere personal idiosyncracy, pity would appear to me an appropriate moral emotion.
Mercy and pity
Supererogatory mercy does not require pity but we have already seen that punitive leniency precisely does: the relevant concern to alleviate my condition or to contribute to my well-being derives from pity at deserved bad fortune, bad fortune that I have brought on myself.
Shawn Floyd, 'Aquinas and the Obligations of Mercy', The Journal of Religious Ethics, Sep., 2009, Vol. 37, No. 3 (Sep., 2009), pp. 449-471.
Kristján Kristjánsson, 'Pity: a mitigated defence', Canadian Journal of Philosophy , June–August 2014, Vol. 44, No. 3/4 (June–August 2014), pp. 343-364. (The author regards pain at undeserved bad fortune as 'compassion' and not 'pity' but I do not think ordinary usage supports this distinction.)