Is there any proof in the Sunnah that there is a prayer on the last Friday in Ramadan in expiation for the sin of one who missed the obligatory prayers?

Firstly: 

Whoever omits an obligatory prayer until the
time for it has ended, one of the following must be the case: either he
omitted it for a legitimate reason such as falling asleep or forgetting, or
he omitted it without any excuse. If a person omits it with for legitimate
reason, there is no sin on him, but he must offer it when he wakes up or
remembers. If a person omits it deliberately, then he has committed a grave
sin and he must make it up according to many scholars; others favoured the
view that he cannot make it up, but he is required to repent, pray for
forgiveness, regret it, and do a lot of righteous deeds. 

Please see the answer to question no.
13664 

Secondly: 

The report that speaks of a prayer that may
be offered by one who omitted a prayer deliberately until the time for it
ended, so that it may be an expiation for his deeds, is a lie and is falsely
attributed to Islamic teaching. There follow some comments of the scholars
concerning that: 

1.    
Ash-Shawkaani (may Allah have mercy on him) said: 

The hadeeth “Whoever prays the five
obligatory prayers of the night and day on the last Friday of Ramadan, that
will make up for what he missed of prayers during that year” is undoubtedly
mawdoo‘ (fabricated). I could not find it in any of the books in which the
authors compiled fabricated hadeeths, but it is well-known among a group of
people who claim to have knowledge in the city of San‘aa’ nowadays, and many
of them do that! I do not know who fabricated it and told it to them; may
Allah curse the liars.

End quote from al-Ahaadeeth al-Mawdoo‘ah
(p. 54) 

2.    
The scholars of the Standing Committee for Issuing Fatwas said: 

Prayer is an act of worship and the basic
principle concerning it is tawqeef i.e., stopping at what has been narrated
in sharee‘ah (and not introducing anything new); the issue of making up the
prayers and details of how to do that is something to be learned from
textsand it is not valid to refer to anything in this regard except the Book
of Allah, the Sunnah of His Messenger (blessings and peace of Allah be upon
him) and the consensus based on both of them or one of them. There is no
proof from the Prophet (blessings and peace of Allah be upon him), or from
his Companions (may Allah be pleased with them), or from the leaders of
guidance (may Allah have mercy on them) that they offered this prayer or
instructed or encouraged others to do so. If it was something established,
then his Companions (may Allah be pleased with them) would have known about
it and would have transmitted it to us, and the leaders of guidance after
them would have guided us to it. But there is no proof to that effect from
any of them in word or in deed. That indicates that what is mentioned in the
question about the “prayer to make up for the prayers of a lifetime (salaat
al-qada’ al-‘umri)” is a bid‘ah that has been introduced into Islam for
which Allah has not prescribed. It is proven from the Prophet (blessings and
peace of Allah be upon him) that he said: “Whoever introduces anything into
this matter of ours that is not part of it will have it rejected.” (Agreed
upon). Rather what the Messenger of Allah (blessings and peace of Allah be
upon him) instructed us to do is to make up the prayers that one misses
because of falling asleep or forgetting until the time for them ends, and he
explained to us that we should offer those prayers as they are usually
offered when we wake up or remember, not on the last Friday of Ramadan.

End quote from Fataawa al-Lajnah ad-Daa’imah
(8/167, 168) 

Shaykh ‘Abd al-‘Azeez ibn Baaz, Shaykh ‘Abd
ar-Razzaaq ‘Afeefi, Shaykh ‘Abdullah ibn Ghadyaan, Shaykh ‘Abdullah ibn
Qa‘ood 

3.    
Shaykh Muhammad ibn Saalih al-‘Uthaymeen (may Allah have mercy on
him) was asked: 

There is a group of people who have a custom
in Ramadan, which is to offer the five obligatory prayers after a prayer on
the last Friday (of Ramadan), and they say that this is to make up for any
obligatory prayers that a person did not offer or that he forgot in Ramadan.
What is the ruling on this prayer? 

He replied: 

The ruling concerning this prayer is that it
is an innovation (bid‘ah) and it has no basis in Islamic teachings; it will
only take a person further away from his Lord, because the Messenger of
Allah (blessings and peace of Allah be upon him) said: “Every innovation is
misguidance, and every misguidance will be in the Fire.” No matter how much
the followers of an innovation like it and regard it as good in their own
minds, it is something bad before Allah, may He be glorified and exalted,
because His Prophet (blessings and peace of Allah be upon him) said: “Every
innovation is misguidance, and every misguidance will be in the Fire.” These
five daily prayers that a person makes up on the last Friday of Ramadan have
no basis in Islamic teaching. Moreover, we say: has this person missed only
five prayers? Perhaps he missed a number of days, not a number of prayers? 

What matters is that so long as a person is
aware that he has missed prayers, he must make them up when he realises
that, because the Prophet (blessings and peace of Allah be upon him) said:
“Whoever sleeps and misses a prayer or forgets it, let him off it when he
remembers it” (Agreed upon). But if a person does these five prayers to be
on the safe side – as they claim – that is something reprehensible and is
not permitted.

End quote from Majmoo‘ Fataawa ash-Shaykh
al-‘Uthaymeen (12/227, 228) 

4.    
Shaykh Saalih al-Fawzaan (may Allah preserve him) was asked: 

I read a hadeeth from the Messenger
(blessings and peace of Allah be upon him) in which he said: “Whoever misses
a prayer during his life and cannot remember what it was, let him get up on
the last Friday of Ramadan and pray four rak‘ahs with one tashahhud,
reciting in each rak‘ah the Opening of the Book (i.e., al-Faatihah) and
Soorat al-Qadr fifteen times, and Soorat al-Kawthar the same number of
times, and let him say in his intention: I intend to pray four rak‘ahs as
expiation for what I have missed of prayers.” How sound is this hadeeth? 

He replied: 

There is no basis for this in the Sunnah of
the Messenger (blessings and peace of Allah be upon him). What is proven
from the Messenger (blessings and peace of Allah be upon him) is that he
said: “Whoever forgets a prayer or sleeps and misses it, let him offer it
when he remembers it. There is no expiation for that except this” (agreed
upon). With regard to the prayers that you missed previously, if you missed
them because of sleep – for example – or because you were unconscious, or
for some excuse that you thought made it permissible for you to delay them,
then what you must do is make them up and offer them in order. If you missed
them deliberately, then the more correct of the two scholarly views is that
you must repent to Allah, because the one who misses prayer deliberately is
in a grave position, even if he does not deny that they are obligatory. The
correct view is that he has become a disbeliever by doing that. So you must
repent to Allah if you missed them deliberately, and you must persist in
offering prayers regularly in the future, and may Allah accept your
repentance. 

But if you missed them because of sleep or
unconsciousness, or something else that prevented you from offering them on
time, then you have no choice but to make them up. As for offering this
prayer that you mentioned at the end of Ramadan in this manner, there is no
basis for that in Islamic religion, and it will not make up for the prayers
that you missed. End quote. 

For more information please see the answer
to question no. 49612 

And Allah knows best.