Right at the beginning of the Tur and [Shulchan Aruch, siman 1][1], it says that it is better to pray a little with kavannah (proper concentration?) than more without. I don't know that anyone argues. I also don't see it put into practice in many places where I might have applied it.  
The Rambam ([Hil. Tefillah 4(14)][2]) says pretty clearly that we should not daven at all without kavannah, and if we did, go back and pray again. But, later authorities seem to tone it down - after all, we don't have so much kavannah anyhow.  
That is pretty much the answer I have gotten when I asked, say: I'm really tired. Maybe I should sleep instead of going to shul for maariv, and hope I get up in time to daven later. No, no! Go to shul now and do the best you can.  
Maybe I should say less of Pesukei D'zimrah? No, no! Birkas Krias Shema - are you nuts?    
Anyhow, when Slichos comes around I really struggle. Do I have kavannah for more, or am I just stretching the kavannah I have much thinner? By the time I get to Hodu I'm frequently exhausted [I do have health issues with fatigue]. What should one do? "*How* can you think of skipping prayers at a critical time like the Aseres Yemei Teshuvah?!"  
Also, what if a person can follow with the words, and pays attention, but is past the point of *enjoying* what's happening? He might have felt more *inspired* if he had said less. Does that play a role?  
I'm asking for sources for practical rulings, but also advice on the right attitude.  

Update: On re-reading, I don't want the impression that the question is based on my physical situation, as if I'm tired all the time. The real question is more based on a *mental* weariness, on one's ability to "hold kop", the point where one feels prayers turning into mechanical reading without really paying attention.  
Update: tesvov has suggested in a number of comments that the principle stated by the Tur *only* applies to optional additions, like the Tikkun Chatzos that is the topic where he states it. Interesting suggestion - obviously Rav Avigdor Miller quoted below disagrees - and it would be good to have an explicit statement by a posek about it. [ \[Berachos 34a][3] might be a good source on this, but of course in those days prayers were more fluid.]  
Third update: My son-in-law suggested what may be a compromise position, though personally I find it rather a depressing one: The requirement for kavannah is indeed absolute, and according to the [Rambam][2] one should perhaps never say any part of the davening, even the Amidah, without proper kavannah. *However*, since in our times we don't have so much kavannah anyhow, this no longer serves as an excuse to skip anything.  
You really have to see the wording of the Rambam there - not really quoted by later poskim, but super-powerful and perfectly clear [ital. mine].  

> A person finds that his thoughts are confused and his mind is
> distracted: He *may not pray* till he has recovered his mental
> composure. Hence, on returning from a journey or if one is weary or
> distressed, it is forbidden to pray till his mind is composed. The
> sages said that he *should wait three days* till he is rested and his
> mind is calm, and then he recites the prayers.


  [1]: https://www.sefaria.org/Shulchan_Arukh%2C_Orach_Chayim.1.4?lang=bi&with=all&lang2=en
  [2]: https://www.sefaria.org/Mishneh_Torah%2C_Prayer_and_the_Priestly_Blessing.4.15?lang=bi&with=all&lang2=en
  [3]: https://www.sefaria.org/Berakhot.34a.11?lang=bi&with=all&lang2=en