If a man wants God to accept his supplication, he should ask for the softness of the heart and work for it. When a heart is soft, veils between a man and God will be lifted and he will be closer to God.
The method of question and supplication has an effect in softening the heart. Hadiths talking about humiliation while supplication were mentioned to prove this goal.
Ahmad bin Fahed Al Houli in Oddat Al Da’i Book (the supplicator appliances book) says that when God’s prophet (pbuh) supplicates, he behaves the same as when he feeds the poor. 
It was narrated that among what was divinely revealed to Moses (a.s):
“When you ask me, open your hands and supplicate with humiliation like a servant who cries for the help of his Lord. If you do so, you get mercy, I am the most generous and the able”. 
Mohammad bin Mislim said:” I asked abi Jaafar (a.s) about God’s almighty saying:(... but they did not humble themselves before their Lord, nor would they supplicate (to Him for mercy), so he (a.s) said:” Humbleness is obedience and supplication is raising hands and asking”. 
Because the purpose of this method of supplication was not clear to people, the skeptics used to make people in doubt regarding the method of supplication. Why do we raise our hands to the sky? Will God be on the sky side until we raise our hands to the sky? The imams of Ahl Al Bayt (pbuh), made it clear to them that God Almighty is everywhere, but in this way of supplication we take the slogan of submission and need before God, for raising the hands is a sign of submission and need. This slogan has a suggestive effect in softening the heart, removing hardness from it, purifying it and preparing it in the hands of the Blessed and Exalted.
Al Tabarsi reported in Al Ihtijaj that abo Qorra said to Al Imam Al Rida (a.s)” Why do you raise your hands to the sky when you supplicate? Abo Al Hasan (a.s) answered: God has enslaved his creations with a variety of worship… and enslaved them when they supplicate and invoke God by opening hands and raising them to the sky to show submissiveness as a sign of slavery and humiliation to Him”. 
The moments of softness are the moments of mercy descending. A man should exploit these moments to head to God by supplication because mercy descends in these moments without an account. This is not because God’s mercy has a specific and limited time, but there is a specific time and specific case to receive this mercy. It is the case of heart softness since when a man’s heart is softened, it can receive this mercy.
The prophet Mohammad (pbuh) says:” Take advantage of supplication in the time of heart softness because it is the time of mercy”. 
Abo Basir says that Abi Abdullah Al Sadek (a.s) says:” If you feel your heart is softened, supplicate because a heart is not soft until he’s sincere”. 
Abo Abdullah (a.s) says:” If you have gooseflesh and your eyes are filled with tears, supplicate for your needs and you’ll be answered”. Abo Abdullah (a.s) says:” If you have gooseflesh, your eyes are filled with tears and your heart is sacred, supplicate for your needs and you’ll be answered”. 
The hadith is precise because answering supplication is directly related to the supplicator’s status. The more the heart is softened, the supplicator will be more answered. On the other hand, the more the heart is bitter, the more distant the supplication will be.
It was mentioned in the islamic texts to benefit from the moments of broken heart and heart softness when a man gathers his misfortunes and sorrow in this life to head to God almighty by supplication. These moments prepare a man to head to God and get his mercy. The secret behind that is that the heart cannot head to God and receive from Him except at the time of heart softness.
Then a man who wants to move towards God’s face and head to him in supplication should get this softness.
Ishak bin Ammar says:” I said to Abi Abdullah (a.s): When I supplicate, I wish to cry, but I don’t… but when I remembered the death of my family members, I feel my heart soft and cry. Is it permissible? So he said: yes, when you feel your heart soft cry and supplicate for your God, exalted. 
If he is not able to cry to feel the softness of his heart, he can pretend weeping because it leads to crying, and the heart's softness makes the heart open towards God almighty. Saed bin Yasar says:” I said to Abi Abdullah Al Sadek (a.s): I pretend weeping while supplicating, but I can’t cry. So he said “Yes, you can do that”. 
Abi Hamza says:” Abo Abdullah (a.s) says to Abi Basir: If you fear something or you have a need, start with the remembrance of God, glorify Him, praise Him, send blessings to the prophet Mohammad and his family, then ask for your need, and pretend crying… my father used to say: ``A servant is no closer to God than when he’s prostrating and crying”. 
Supplication of Ahl Al Bayt (peace be upon them), Al Sheikh Mihammad Mahdi Al Asdi (may God send mercy on him)
 Oddat Al Da’i 139, Al MajalesBook for Al Mofeed 22
 Oddat Al Da’i 139
 Surah Al Mo’men, verse 76
 Usul Al Kafi 2: 348.
 Usul Al Kafi: 522, Wasa’il Al Shi’ah 4: 1101, Volume: 8687.
 Bihar Al-Anwar 93: 313.
 Wasa’il Al Shi’ah 4: 1120, H: 8791, Usul Al Kafi
 Wasa’il Al Shi’ah 4: 1 114, H: 8763.
 Wasa’il Al-Shi’ah 4: 1 4 11, H: 8766.
 Usul Al Kaf: 523, Wasa’il Al Shi’ah 4: 1121, AH: 8767.
 Wasa’il Al Shi’ah 4: 1122, AH: 8769, Usul al-Kafi: 523.
 Usul Al Kafi: 452, Wasa’il Al Shi’ah 4: 1122, Volume: 8770.