Why and how to call upon Allah in my prayer?.

     And your Lord says: "call upon Me, I will respond to you, indeed, those who desdain My worship will enter hell contemptible "disgraced" (ghafir:60). And He says: "when My servants ask you O Muhammad concerning Me, indeed, I am near, I respond to the invocation of the supplicant when he call upon Me, so let them respond to Me (by obedience), and believe in Me that they may be rightly guided (Albaqarah:186).

     Here Allah, Almighty, says call upon Me I will answer you, and our prophet (pbuh) said: "the supplication is the worship", and those who are too proud to call on Allah Azza wa Galla and single Allah out will surely enter hell in humiliation.

     As for the acceptance of supplication (dua'a) it takes one of the three forms given here:

1- that one gets exactly what he asked for.

2- that in lieu of what one wanted to have, one was given a certain return or reward of the hereafter.

3- that one did not though get what he wanted but some hardship or calamity that was due to fall on him stood removed.

And about how to call on Allah Azza wa Galla in the prayer, there are many supplications from the Sunnah in every pillar of the prayer, here I will write some supplications in Arabic and the meaning of it in English you can simply recite it in your prayers.

1- At the beginning of the prayer:

At the beginning of the prayer only in the first rakhah of any prayer, between Allahu Akbbar and reciting Alfatihah say:"Allahumma ba'ed bayni wa baina khatayaia kama ba'adta bayna almashreq wa almaghreb, Allahumma naqini min khatayaia kama yuonaqa althawbu alabiadd mina aldanas, Allahumma eghselni min khatayaia bethalag wa alma'e wa albarad, Allahu Akbbaru kabera wa alhamdu lellahi kathera wa subhana Allahi bukratan wa asila".

its meaning: 

     O Allah separate me from my sins as you have separated the east from the west, O Allah cleanse me of my transgressions as the white garment is cleansed of stains, O Allah wash away my sins with ice and water and frost, Allah is the greatest, most great, praise is to Allah, abundantly, glory is to Allah at the break of day and at its end.

2- During ruku': 

 During ruku' say: subhana rabbia aladheem (three times), subbuoh quodos rabbu almala'ekati wa alruoh.

its meaning: 

 glory to my Lord the exalted, glory to You, most holy are you, Lord of the angels and the spirit.

3- On rinsing from the ruku':

On rinsing from the ruku' say: same'aa Allahu le-man hamedah.

its meaning:

Allah hears whoever praises Him.

4- After rinsing from ruku':

Rabbana wa laka alhamdu hamdan katheran tayiban mubarakan feeh kama yanbaghi legalali wajheka wa adhimi sultanek.

its meaning:

Our Lord praise is yours, abundant, good and blessed praise which suit your glory and your greatness of dominance.

5- In prostration: 

Subhana rabbia al'ala (three times), sajada wajhi lellazi khalaqahu wa shaqa sam'ahu wa basarahu bihawlihi wa qwatihi fa-tabaraka Allahu ahsanu alkhaleqeen.

its meaning:

Glory be to my lord, the most high, I have prostrated my face on the one who created it, and gave it hearing and sight by His might and His power, glory is to Allah, the best of creators, then you can call on Allah for what you wanted.

6- Between two prostration:

Rabbi eghferli , Rabbi eghferli.

its meaning:

My Lord forgive me, my Lord forgive me.

7- After final tashahuod:

When ending the prayer before Alssalam say: Allahumma inee a'auzu beka min azabi alqabr, wa min azabi jahanam, wa min fetnati almahyia wa almamat, wa min sharri fetnati almaseeh aldajaal.

its meaning:

O Allah, I seek refuge in you from the punishment of the grave, and from the punishment of the hell-fire, and from the trails of life and death, and from the evil of the trail of the false messiah.