I applied for a UK Visit visa and indicated that I would be staying
for 1 week due to university commitments. After that, I realised I
could spend one month instead of one week, What can I do to fix that?
There is nothing to 'fix', the 'standard visitor visa' is issued for a term of six months and very rarely (if ever) is this changed. I haven't seen one issued for less than 6 months since the visa was invented in 2015. They like to keep it 'standard' because it reduces the overhead costs.
For a 1-week visa application, is it going to be exactly one week or a
little bit more to allow me to stay longer?
The only time they issue 1 week visas is when the person needs to testify in court or when the person has a performance engagement under the Tier 5 programme. These do not apply to you.
Some things to consider...
You may be asked to present your accommodation arrangements or proof
of onward travel as part of your landing interview;
If the change in your itinerary is material, you will need to advise
the Immigration Officer during your landing interview. Failure to do
that can lead to serious problems even if the IO did not bring it up;
When you ultimately leave the UK, your carrier will give UKVI all of
the details about your exit;