When you have an embedded clause, it could either be an embedded TP or an embedded CP. So, when do you have which?
First, if the embedded clause is finite, you have a CP, that much is guaranteed. Even if you can’t hear the C (since C for finite clauses can either be that or ∅). One example: Pat said [CP (that) [TP Chris left ] ].
If the embedded clause is non-finite, the answer is “it depends.” Sometimes you can hear the C, in which case, of course, there’s a CP. For example: Pat wants [CP for [TP Chris to leave ] ].
Another case where a non-finite clause is a CP is when its subject is PRO. If you have a PRO in an infinitive to-clause, there has to be a CP in order for PRO to get case. For example: Pat wants [CP ∅NULL [TP PRO to leave ] ].
When the subject of the embedded non-finite clause gets case from the higher verb (ECM), then there is no CP, just a TP. For example: Pat wants [TP me to leave ].
Those are, I think, the basic cases. So, it comes down to deciding whether there is a PRO in the embedded clause or not, to a large extent.