The words more powerful than "ask", but less powerful than "demand"

Request

noun: an act of asking politely or formally for something.
verb: politely or formally ask for.

Source: Oxford

It is more formal than 'ask' (say something in order to obtain an answer or some information.), but has less force than 'demand' (an insistent and peremptory request, made as if by right.)

Urge

'Urge' is weaker than 'demand' but is stronger than 'ask'.

I urge you to do this for me.

Instead of asking someone to do something, you can assign the task to do something. More casually, you can "task them to do something".

This has the general feeling of an involuntary obligation (someone else decided you must do it), but also not having serious consequences without additional context to quantify the penalties for failure.

Some examples:

• The professor assigned us reading for next week.
• My manager assigned me the task of coming up with five new product features.
• I've been tasked with drawing up the agenda for the next meeting.

Another word that can be used would be expect. From Dictionary.com in their section on synonyms:

Expect, anticipate, hope, await all imply looking to some future event. Expect implies confidently believing, usually for good reasons, that an event will occur: to expect a visit from a friend.

This can be used both from a superior position (a boss asking strongly that a subordinate do something):

I expect all employees to complete their time-sheets for the previous week by the following Monday afternoon.

And can also be used more formally from a slightly subordinate position or when dealing with a large organisation and you are trying to "force their hand" without (at this stage) sounding too demanding:

Please find enclosed the details of my insurance claim. I expect to receive your initial response within 14 days.

By telling someone of your expectation that a certain event or action will happen, you are, in effect, requesting them to make sure that that it does happen -- more strongly than simply asking them to do something, but not as strongly as demanding that it happen.

A milder form of this idea -- setting an expectation -- would be look forward: "I look forward to receiving your response within 14 days.".

If this is someone telling a subordinate to do something, perhaps "instruct", "direct" or even "tell"?