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This sentence sounds pretty awkward to me but I can't come up with an explanation. Could someone explain it?

I wish to assure you that I should make every effort to be worthy of
the confidence you may place in me.

Another sentence:

It is reported that the railway will be completed by the end of next
month.

It sounds perfect to me, but someone told me that the right version should be "will have been" in this sentence, so is the use of “will be” grammatically wrong here?

You've asked two separate questions, and it is better to ask just one at a time; but I will attempt to answer both.

First, that use of should is odd for most people. There is an old-fashioned pattern of using shall instead of will, and should instead of would when the subject is first person ('I' or 'we'). Some people still use these most of the time; some people rarely use them at all.

Often it depends on the context; so for myself, I would always ask a question with Shall I? not Will I? (unless I am asking for a prediction). I often, but not always, say I shall ... rather than I will. (Of course in most conversation I say I'll, which neutralises the difference). But I rarely say I should for I would, except in the frozen phrase I should think... (Of course there is another meaning for I should = I ought to, and I use that readily).

My point here is that you example seems to be said by somebody who does use I should in that case. For such a speaker, the sentence is normal; but most people today would say I would there.

There is also the question of the tense: should or would there makes it hypothetical: I think most people would choose shall or will, to make it more definite. Even though the may in the last clause does strictly make it hypothetical, most people in that sort of situation would I think choose to be more positive, and state their intention with will or shall.

For the second question: the future perfect is not much used in conversation, unless it is needed for clarity. It would not be wrong to use it, but most people would say will be completed.

I wish to assure you that I should make every effort to be worthy of the confidence you may place in me.

The sentence doesn't seem quite right since:

• wish is used for expressing a strong desire for things that have a low probability to happen. In your case, you are just assuring that person, and there is nothing impossible in that.




• should make: You may want to use a more confirming and reassuring verb phrase. See the first definition of should.




• every effort: If you want to make some effort in proving your reliability to that person, for example, then that is fine. However, it won't be proper to say that in front of that person since it may indicate that you actually have to make efforts in order to be a better version of yourself.




• may sounds correct, but it depends on what you really intended by using it. I find that would makes a better candidate as it is used to indicate something that may or may not happen. See would definition as a modal verb when used in probabilities.




• place is correct too, but there are better and more common words to use when talking about "confidence in someone".

Therefore, I would restate your sentence like this:

I want to assure you that I will be worth the confidence you would have in me.

I wish to assure you that I should make every effort to be worthy of
the confidence you may place in me.

I don't quite agree with the other answers.

My answer

You need to match tenses.

I wish to assure you that I shall make every effort ... (Both verbs are simple present)

I wished to assure you that I should make every effort ... (Both verbs are simple past. They describe what has already happened.)

If you say, "I wish to assure you that I should make every effort ..." then you are indicating an obligation (that is also hypothetical) on your part.

Note: The railway sentence is perfectly correct.

Subjunctive mood.
https://en.wikipedia.org/wiki/English_subjunctive

Answering the first one: "should" here is a modal, indicating the subjunctive mood. This mood indicated that the sentence refers to a wish or hypothetical situation, not actual reality. It is used appropriately here as the sentence describes the speaker's wish.

The subjunctive is more often used in other languages and less and less in English, where its use can seem archaic (and can be useful in that role; for example marking a character's English as archaic). For example "I wish it were" is the subjunctive and correct as describing a wish, but a lot of people would say "I wish it was".

As regards should in particular:

The English modal verbs do not have present subjunctive forms. A
construction with the modal should is frequently used as an
alternative to the simple present subjunctive, e.g., It is important
that he should be cured.
my emphasis

In the sentence given here instead of be, the verb is make, and the addition of should indicates the subjunctive mood.

I have nothing for "completed".

ADDENDUM for @chasly from UK

An example: from Shakespeare's Antony and Cleopatra (1185)

O, that his fault should make a knave of thee,

The hypothetical circumstance (O, that instead of I wish) and same construction using should and make.