Can a Muslim fast and pray for the purpose of supplication?

The other question is can one go to the Al Haram to pray for one's
needs, e.g to pray for cure from sickness , to pray to have children
etc. Still can one fast for some days for the purpose of requesting
for one's needs from Allah for the reasons stated above? (i.e. to pray
for children etc).
Praise be to Allaah.
There is no reason why aMuslim should not pray in al-Masjid al-Haraam
for the purpose of supplication, but it is better for him not to
restrict his intention in the prayer to supplication. Rather his
intention should be to worship Allaah through this prayer and to hope
for its reward in the Hereafter. Moreover, prayer includes dhikr,
reciting Qur'aan, bowing, prostrating anddu'aa', so the du'aa' should
just be a part of the prayer; it is not the primary purpose. Al-Masjid
al-Haraam is one of the blessed and venerated places, so if a person
prays and calls upon Allaah whilst prostrating, for example – then he
will have combined the virtue of the place with the virtueof the
action. If this is done during the last third of the night, he will
also add the virtue of the time.
Shaykh al-Islam Ibn Taymiyah said:
With regard to the words of the Prophet (peace and blessings of Allaah
be upon him), "As for rukoo', glorify your Lord therein, and as for
sujood (prostration) strive in du'aa' therein, because then it is more
likely that you will be answered" (narrated by Muslim, 479, from the
hadeeth of Ibn 'Abbaas) – this contains the command to glorify Allaah
in rukoo', and the command to make du'aa' when prostrating indicates
that du'aa' in sujood is more likely to be answered than in rukoo'.
Hence he said, "because then it is more likely that you will be
answered". And he said, "The closest that the slave is to his Lord is
when he is prostrating." (Narrated by Muslim 482 from the hadeeth of
AbuHurayrah). This is a command that du'aa' should be made in sujood.
Majmoo' al-Fataawa, 22/378
And he said:
Du'aa' is answered whenrain is falling, when battle in joined, when
the adhaan and iqaamah are given, following prayers, when prostrating,
and when offered by one who is fasting, one who is travelling and one
who is oppressed, and so on. All of this is narrated in well-known
ahaadeeth in the books of Saheeh and Sunan. Du'aa' is also likely to
be answered in the sacred places such as'Arafah, Muzdalifah, Mina, the
Multazim and other sacred places in Makkah, and in the mosques in
general. The greater the virtue of the mosque – such as the three
Sacred Mosques [inMakkah, Madeenah and Jerusalem] – the better prayers
and du'aa's are in them.
Majmoo' al-Fataawa, 27/129-130
On the other hand it maybe said:
Doing righteous deeds in the hope that du'aa' will be answered is
something that is prescribed in sharee'ah. Hence doing wudoo' and
praying in al-Masjid al-Haraam may be included among the righteous
deeds that may be done before starting du'aa' in the hope that that
the du'aa'will be answered.
It was narrated from 'Uthmaan ibn Hunayf that a blind man came tothe
Prophet (S) and said:Pray to Allaah to heal me. He said, "If you
wish,you may delay the reward until the Hereafter, for that is better,
or if you wish, I will make du'aa' for you." He said, Make du'aa'. So
he told him to do wudoo' and do it well, and to pray two rak'ahs, and
to say this du'aa': "Allaahumma innias'aluka wa atawajjahu ilayka bi
Muhammadin Nabi al-rahmah. Yaa Muhammad inni qad tawajjahtu bika ila
rabbiy fi haajati haadhihi li tuqdaa. Allaahumma shaffi'hu fiyya (O
Allaah, I ask of You and I turn my face towards You by the virtue of
Muhammad theProphet of Mercy. O Muhammad, I have turned my face by
virtue of you to my Lord concerning this need of mine so that it may
be met. O Allaah, accept his intercession concerning me)."
Narrated by al-Tirmidhi, 3578; Ibn Maajah, 1385; this hadeeth was
classedas saheeh by al-Tirmidhi and by al-Albaani in Saheeh al-Jaami',
1279
The Prophet (peace and blessings of Allaah be upon him) commanded him
to do wudoo' and pray two rak'ahs before he made du'aa'; this
indicates that prayer is one of the righteous deeds by virtue of
whicha person's du'aa' may bemore likely to be answered, and that may
be one of the causes of the du'aa' being answered.
With regard to fasting, what has been said with regard to prayer may
also be said with regard to fasting. The intention behind it should be
to worship Allaah by meansof this great act of worship, and to attain
the reward for it in the Hereafter, and to fear Allaah and earn His
pleasure. Then if he is fasting, it is mustahabb for the fasting
person to make a lot of du'aa', for the du'aa' of the fasting person
will be answered, especially at the time of breaking the fast.
And Allaah knows best.