An alert state of mind is one of the many vital ingredients of Prayer. If you are tired and exhausted, it is better to rest and do some deep breathing and relaxation exercises before you make the commitment to stand in front of your Lord. And yes, it is preferred to pray a bit late but with attention and concentration. Of course, the delay in performing Prayer from its allocated time should only be for the length of time one has to relax and rejuvenate himself, gain energy to stand before the Almighty. As Imam ‘Ali (AS) has advised us:

Do not offer Prayer in the state of drowsiness. While offering Prayer, do not think about yourself because you are standing in front of Allah (SWT). Indeed, only that portion of the Prayer will be accepted that he has paid full attention towards Allah (SWT).

al-Saduq (RA) reports from Imam ‘Ali (AS) that he said:

None of you should ever stand for Prayer in a lazy or drowsy state, nor should you let (random) thoughts pass through your mind (in the state of Prayer). For, in that state, you stand before your Glorious and Almighty Lord. Verily, the reward a devotee derives from Prayer is equal to the extent of it that he offers with an attentive heart.
(al-Hurr al-`Amili, Wasa’il al-Shi’ah, iv, 687)

In Fiqh al-Ridha it is stated:

When you want to perform the Prayer, do not go to it lazily, sleepy, hurriedly, or unmindfully. Approach it calmly, solemnly and slowly. You should display submission, imploringly and humbly to Allah (SWT). You are to show awe and signs of fear and hope, with caution and apprehension. Thus, you are to stand before Him, as an escapee and sinful slave stands at attention before his master; on the alert, with joined heels, erected trunk, not looking to right and left, reckoning as if you see Him. If you do not, He certainly does see you…
(Mustadrak al-Wasa’il, Chapter 1, Tradition 7)

Imam (AS)’s posture always allowed him to relax and to be comfortable. Your posture in Prayer also has to allow you to remain alert and aware. There is a two-way relationship between slumping and distraction. Good posture means there is musculoskeletal balance. This balance helps to protect the joints in the spine from excessive stress. It also guards against injury and possible deformity. Good posture helps prevent distress and pain that lead to distraction.

As your concentration in Prayer progresses, your body metabolism will generally become slower and slower, until your heartbeat and breathing are barely perceptible. At this point, when you are totally focused in your Prayer, it is when you achieve your greatest benefit from the worship that has limitless rewards. You would seem to be at a point where mental fatigue and stress is easily released.

Upon completion of your Prayer, most often you will feel relaxed, renewed, alert, and better focused. Your senses will be more acute. You will often find yourself with a heightened sense of awareness. You will generally feel confident and secure in a positive state of mind. Happiness will be yours. This is the state of mind you should strive to achieve after completion of Prayer.