Cittassa

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Why citassa and not just citta?
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The particular case of Assa:

Assa = opt. of atthi [Sk. syāt >> {opt. ac. sg.} √ अस् as]
√ अस् as
- to become (BṛĀrUp.)

Citta+assa >> the particular use of a noun compounded with a verb.
Usually this is done with verbs like kar (kṛ,) bhū, as, and the sort (with a general meaning of becoming).
In our case a conditional (optative) assa (or (the Pali) siyā) would be added to citta.

Cittassa (citta-assa,) is the existential form of the Citta. 
Particularly in the following extract:
With the origination of name-and-form there is the origination of (the existential) mind. 
Nāmarūpasamudayā cittassa samudayo. 
SN 47.42

We know that, in paṭiccasamuppāda, the first occurence of citta, is due to the origination of ignorance (vijjā); and that it happens in the Saṅkhāra nidanā - see here and also here (the definition of cittasaṅkhāra).
However, SN 47.42 above cited, tells us that the origination of name & form (nāmarūpa), is at the origin of cittassa.
If you read the sketch properly, you will notice that the descent of nāmarūpa in saḷāyatana "creates" a new khandha called cetanā. And this is the existential part of citta (the cittassa - the "becoming" citta). The part that is proper to satta.
It is important to notice that in the Agama (SA 298,) the khandhas in the nāmarūpa nidanā are different than the khandhas given in the Nikaya (SN 12.2). It  is not a discrepancy between the two schools. They both see the khandhas in relation to their position in paṭiccasamuppāda. The Khandhas of the Nikaya are the khandhas that have "become" in satta. They are the existential khandhas of the sphere of senses (saḷāyatana). 
To understand the descent of nāmarūpa in saḷāyatana, see here (establishing of consciousness).

Cittassa is the "Being" form, of the "Cosmic" (immaterial) Citta.
When one's Citta gets liberated, it is from the bound of this existential cittassa.

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With the origination of name-and-form there is the origination of mind.
Nāmarūpasamudayā cittassa samudayo.
SN 47.42

The defiled mind, Visakha,  is cleansed by exertion.
upakkiliṭṭhassa, visākhe, cittassa upakkamena pariyodapanā hoti.
AN 3.70

Dwell contemplating mind in mind … in order to know mind as it has come to be.
citte cittānupassino viharatha ātāpino sampajānā ekodibhūtā vippasannacittā samāhitā ekaggacittā, cittassa
yathābhūtaṃ.
SN 47.4

Because, bhikkhus, that foolish, incompetent, unskilful bhikkhu does not pick up the attribute of his own mind.
Tathā hi so, bhikkhave, bālo abyatto akusalo bhikkhu sakassa cittassa nimittaṃ na uggaṇhāti.
SN 47.8

Here, bhikkhus, the noble disciple gains concentration, gains one-pointedness of mind, having made release the object. This is called the faculty of concentration.
Idha, bhikkhave, ariyasāvako vossaggārammaṇaṃ karitvā labhati samādhiṃ, labhati cittassa ekaggataṃ—idaṃ vuccati, bhikkhave, samādhindriyaṃ.
See Note* below on Vossaggārammaṇa, etc.
Also, it is possible - and most probably - that this sentence encompasses a genitive absolute. In this case, citassa would just be a genitive - (just as samādhiṃ - and ekaggataṃ (the past participle declined to a genitive in such a grammatical construction).
In this case, the translation above would be:
Here, bhikkhus, the noble disciple, having undertaken the relinquishing of the support, gains concentration and gains a citta, caused to become one. This is called the faculty of concentration
SN 48.9

It is, bhikkhus, when concentration by mindfulness of breathing has been developed and cultivated that no shaking or trembling occurs in the body, and no shaking or trembling occurs in the mind.
Ānāpānassatisamādhissa, bhikkhave, bhāvitattā bahulīkatattā neva kāyassa iñjitattaṃ vā hoti phanditattaṃ vā, na cittassa iñjitattaṃ vā hoti phanditattaṃ vā.
SN 54.7

The world is led around by mind;
By mind it’s dragged here and there.
Mind is the one thing that has
All under its control.
Cittena nīyati loko,
cittena parikassati;
Cittassa ekadhammassa,
sabbeva vasamanvagū'ti.
SN 1.62

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* Note from SN 48.9

Vossaggārammaṇa:
(lit. having undertaken the relinquishing of the support).
Vossagga [=ossagga； ava+sṛj]
- relinquishing； handing over.
Ossagga，[fr．ossajati]
Ossajati (ava-sṛj)
- To give up，relinquish，renounce.

Ārammaṇa means a support (Sk: āsthāna) and something to grasp on (Sk: grabhaṇa - √ग्रह् grah - to perceive (with
the organs of sense or with [mánas] ) , observe , recognise RV. VS. ŚBr. MuṇḍUp. ŚvetUp.)

Controversial: आलम्ब् ālamb [ā-√ लम्ब् lamb] OR आरम्भण ārambhaṇa [act. ārabh] √ रभ् rabh [linked to √ लभ् labh]

अव ava - ind.
off , away RV.

√ सृज् sṛj
- to release RV. AV.
- to cause to let loose , let go , create Br.
“
(E.g. abandoning lust for form, implies that the ārammaṇa (the hold you have on the support of) form is cut off -
Māra finds an opportunity and an ārammaṇaṃ (a support and something to hold on) to someone not mindful).

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Also, check this: Mano & Citta 

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