Can We Make Up Missed Prayer after `Asr?
Question

People say that no Qada’ (due) prayer can be offered immediately after the 'Asr' prayer. Please help by giving me a comprehensive answer regarding this saying.
Praise be to Allah.Forbidden prayer times
There are some times at which the Prophet (peace and blessings of Allah be upon him) disallowed prayer . These are: 

From after Fajr prayer until the sun has appeared and risen to the height of a spear, i.e., approximately fifteen minutes. (Ash-Sharh Al-Mumti’, 4/162)
The time when the sun is at its zenith in the middle of the sky. This is a short time before the time for Dhuhr prayer begins, approximately one-quarter of an hour or twenty minutes. (Fatawa Ash-Shaykh Ibn Baz, 11/286). Some of the scholars said that it was shorter than that. Ibn Qasim (may Allah have mercy on him) said that it is a brief time period, not even long enough to pray, only enough to say the Takbir to start the prayer. (Hashiyat Ibn Qasim `ala Ar-Rawd Al-Murbi`, 2/245) 
After `Asr prayer until the sun sets. 

These three times were mentioned in many Hadiths in which the Prophet (peace and blessings of Allah be upon him) forbade praying at these times . These Hadiths include the following: 

Al-Bukhari (586) and Muslim (728) narrated that Abu Sa`id Al-Khudri (may Allah be pleased with him) said: “The Messenger of Allah (peace and blessings of Allah be upon him) said: `There is no prayer after `Asr prayer until the sun has set, and no prayer after Fajr prayer until the sun has risen.’” 
Muslim (832) narrated that `Amr ibn `Abasah As-Sulami (may Allah be pleased with him) said: “I said, `O Prophet of Allah, tell me what Allah has taught you and I do not know, tell me about the prayer.’ The Messenger of Allah (peace and blessings of Allah be upon him) said: `Pray Subh (Fajr) prayer, then do not pray until the sun has appeared and risen… then pray for your prayer is witnessed and attended (by the angels), until the shadow of a spear points due north  – which is the time when the sun is in the middle of the sky – then stop praying, for at that time Hell is stoked up. When the shadow turns – which is the beginning of the time for Dhuhr – then pray, for your prayer is witnessed and attended (by the angels), until you pray `Asr. Then stop praying until the sun sets…” 

Can we make up for a missed prayer after `Asr?
What is meant by making up prayers is doing the prayer after the time for that prayer is over. The made-up prayer may be either obligatory or supererogatory. 

Obligatory prayers: The Muslims must pray regularly at the times defined by Allah for each prayer. Allah says (interpretation of the meaning): 

“Verily, the prayer is enjoined on the believers at fixed hours.” [An-Nisa’ 4:103]
Delaying the prayer until its time is over without a reason or excuse is prohibited, and it is a major sin. 
If the Muslim has an excuse, such as sleeping or forgetting, and he was not able to do the prayer on time, then he must make up the prayer when that reason no longer applies , even if that is at one of the times when prayer is disallowed. This is the view of the majority of scholars. (See Al-Mughni, 2/515) 
The evidence for that is the words of the Prophet (peace and blessings of Allah be upon him): “Whoever sleeps and misses the prayer or forgets it, let him pray it when he remembers .” (Narrated by Al-Bukhari, 597 and Muslim, 684) 

Sunnah prayers : the scholars differed with regard to making them up at times when prayer is disallowed. The correct view is that they should be made up . This is the view of Ash-Shafi`i (may Allah have mercy on him). (See Al-Majmu`, 4/170) It was also the view favoured by Shaykh Al-Islam Ibn Taymiyah (may Allah have mercy on him), as stated in Al-Fatawa, 23/127. This is indicated by a number of Hadiths, such as: 

The report narrated by Al-Bukhari (1233) and Muslim (834) from Umm Salamah (may Allah be pleased with her), that the Prophet (peace and blessings of Allah be upon him) prayed two Rak`ahs after `Asr, and she asked him about that. He said: “Some people from `Abd Al-Qays came to me and distracted me from praying two Rak`ahs after Dhuhr, so I did them now.” 
Ibn Majah (1154) narrated that Qays ibn `Amr said: the Prophet (peace and blessings of Allah be upon him) saw a man praying two Rak`ahs after he had prayed Fajr prayer. The Prophet (peace and blessings of Allah be upon him) said, “Are you praying Fajr twice?” The man said to him, “I did not pray the two Rak`ahs that come before it , so I offered them now.” The Prophet (peace and blessings of Allah be upon him) remained silent. This was classed as authentic by Al-Albani in Sahih Ibn Majah (984). Ibn Qudamah (may Allah have mercy on him) said: the fact that the Prophet (peace and blessings of Allah be upon him) remained silent indicates that it is permissible. (Al-Mughni, 2/532) 
And Allah knows best. May Allah bless our Prophet Muhammad and grant him peace.