Discussion of philosophy
So there's a lot going on here but the message is very simple. Even for a native English speaker, this is some flowery language. Very heavily Christian (all the Thee's and Thy's and Thou's are of course God with that Capital G) and it tries to instill humility before God.
All of this is a fancy way of saying "no matter how much I learn and achieve in life, I must never think I've learned or achieved more than God."
An apologist is someone who makes it their work to defend a position, usually one that is under constant barrage from critics. In this case, proving the existence of a Christian God. Now that we know the context, we can examine the poem itself within that context.
From all my lame defeats and oh! much more
From all the victories that I seemed to score;
From cleverness shot forth on Thy behalf
At which, while angels weep, the audience laugh;
From all my proofs of Thy divinity,
Thou, who wouldst give no sign, deliver me.
Taken in context, it begins to make more sense. Lewis is reflecting on his attempts at refuting critics of Christianity. Sometimes he fails, sometimes he succeeds. The last verse is a call to God to help out his cause, to provide solid proof of His existence, yet He does not.
Thoughts are but coins. Let me not trust, instead
Of Thee, their thin-worn image of Thy head.
From all my thoughts, even from my thoughts of Thee,
O thou fair Silence, fall, and set me free.
Lord of the narrow gate and the needle’s eye,
Take from me all my trumpery lest I die.
The second verse is a bit more esoteric. Taken as a whole, it is not so much a confession of defeat, although it makes use of defeatist language, but an admission that at the heart of Christianity is faith. Faith within religion is a complete trust regardless of the existence of material proof. "Let me not trust, instead Of Thee", "O thou fair Silence, fall, and set me free.". The fifth verse is a reference to Bible verses, and the last verse is a call to deliver them from their own mortal cleverness in defending their position.
This is my own interpretation and could likely be completely wrong, so take from it what you will.
I think the core meaning is that his attempt to understand and explain something intellectually (in this case, God) has interfered with his ability to understand it intuitively or emotionally.
Christian prayers for a specific profession or personal inclination are designed to highlight "besetting sins" (meaning, basically, bad habits) that such a person is likely to encounter in their regular activities, and the apologist's is no different.
In discussing this, I'm going to use he/him to refer to the speaker/writer of the poem, for ease. I would use ungendered pronouns, but then I'd be assuming that Christian apology is not a male-dominated field.
In his efforts to rebut criticism of Christianity, the apologist asks for relief:
From all my lame defeats and oh! much more
From all the victories that I seemed to score;
This expresses a desire to be free of the embarassment and frustration of a poor debate performance, and also especially of pride (the first of all sins) in a (perceived, at least) rhetorical victory. It can also express wanting to avoid dwelling on a tally of wins and losses, and to focus instead on his true goal of showing his faith.
From cleverness shot forth on Thy behalf
At which, while angels weep, the audience laugh;
The pray-er wants to win audiences' hearts, minds, and souls with earnest and honest Christian expression, not "trick" them with clever wordplay or rhetorical sleight-of-hand, and asks that the temptation to indulge in (or "shoot forth" i.e. impulsively throw out) such be removed.
From all my proofs of Thy divinity,
Thou, who wouldst give no sign, deliver me.
In a similar vein, the apologist is asking to resist arguing in empirical (scientific or logical) terms (i.e. "proofs"), because he believes that an earnest faith is more valuable than sense-evidence that might undermine it.
Thoughts are but coins. Let me not trust, instead
Of Thee, their thin-worn image of Thy head.
Here the speaker analogizes his distracting thoughts to currency. He sees potential idolatry (temptation to pursue a thing rather than the meaning behind the thing) and a reductive image of his faith on the coin's eroded face, which he rejects in favor of a divine ideal.
From all my thoughts, even from my thoughts of Thee,
O thou fair Silence, fall, and set me free.
Similar to the last couplet, the speaker is asking that thoughts that might inflict some diversion be removed. Like sense-data in the third couplet the apologist is doubtful of his own thinking, and doesn't trust his wayward mind to convey his message well.
Lord of the narrow gate and the needle’s eye,
Take from me all my trumpery lest I die.
The "narrow gate" references The Gospel of Matthew 7:13-14 (KJV) --
13 Enter ye in at the strait gate: for wide is the gate, and broad is the way, that leadeth to destruction, and many there be which go in thereat: 14 Because strait is the gate, and narrow is the way, which leadeth unto life, and few there be that find it.
The needle's eye is a metaphor that Jesus employs in The Gospel of Luke 18:25 (KJV) --
25 For it is easier for a camel to go through a needle's eye, than for a rich man to enter into the kingdom of God.
In referring to these two passages, the speaker is admitting that the pursuit of Christian faith is difficult (in the sense of frustrating, arduous), and asks for conviction and strength to follow his desired narrow path towards heaven. In pursuit of that goal, and of pointing others towards it, he asks that "all my trumpery" be taken away. 'Trumpery' has many senses, but all of them connote worthlessness, referring variably to that which is fraudulent, gaudy, showy, meaningless, nonsensical. The speaker is wishing away any pretense, any subconscious desire to deal dishonestly with his audience.
To sum up, the poem is a request from the speaker that he remain honest and focused on his goal of spreading "the good word". He wishes to be free of all deceit, including self-deceit. One can absolutely identify a deep insecurity in this, but that would be going beyond the text.