Trying to understand what mileage I can claim

My company's office is considered my permanent place of work, which is approximately 120 miles from me. When travelling to an airport, or customer site we use a milage calculator when making claims that takes off the normal commute distance. If i stay at a hotel next to the office, and travel from there to customer sites and return there at the end of the day, is that considered business mileage, or do I still have to deduct the 120 miles that would be my normal commute?

Hi

If the company office is the customers permanent workplace then any travel from their home to that workplace is normal commuting and so not an allowable expense, as per EIM32356, which is what the customer has stated. 
The actual rules about what are classed as ordinary commuting and what is an allowable business journey for the purposes of the employment are a little complex and include the concept of temporary workplaces. Based on what you haves said, I suspect the fact the company deduct 120 miles from every journey that starts at home is administrative ease for them to ensure they keep themselves within the rules for paying travel expenses.
The guidance on Employee travel expenses in general is at EIM31800 onwards. If the individual chooses to stay somewhere nearer to their permanent workplace for ease then any travel from their normal home to get there would not be allowable as it is a private journey to put them in a position to carry out their employment, not actually as part of the duties of the employment. 
Your question really relates to what happens from here. Any subsequent travel from the hotel where they have based themselves would be subject to the normal rules regarding travel expenses, with the hotel in that case effectively being your home for that journey.
If you then travelled to their permanent workplace from there to carry out substantial duties then that journey would also be ordinary commuting (see EIM32055 for what constitutes ordinary commuting).
Any onward travel from the permanent workplace to clients (which are effectively classed as temporary workplaces whilst they are there) would then be allowable business travel, see EIM32230.
If the individual travels from the hotel where they are staying direct to a client (or just briefly stops at the permanent workplace to pick something up for example), then the whole journey may be an allowable expense if the journey is longer (or in a different direction) than what their commute to the office would be from the hotel, and any subsequent travel between clients would be allowable on the basis it's between 'temporary workplaces'.
For tax purposes, HMRC would not require 120 miles to be deducted from the journey as 120 miles is not their 'ordinary' commute if they are staying at a hotel near to the permanent workplace.
This is guidance at EIM32300 and there are some useful examples there too. Ultimately each case would be dependent on the individual facts.

Thank you