The VAT position is really very straightforward. Tax is due on the full value of expenses charged as part of the consideration for a single supply of services (assuming the place of supply is the UK). The liability of the expenses follows the liability of the main supply so if the place of supply is outside the UK and the services are outside the scope of UK VAT then the same would be true of the expenses.
It is a matter of contract what expenses are charged. There is no tax position on the type or value of expenses charged. The only rule is that VAT is charged on the full amount. So a client can be billed for expenses net of VAT where input tax can be reclaimed but equally they can be billed for the gross value including VAT and output tax will be due on whichever amount is billed. Most passenger transport is zero-rated but where travel tickets are charged to clients as expenses VAT will be due on these as well and the same for exempt postage charges.
There is a risk in only charging expenses on the net amount on the basis that the input tax can be reclaimed by the business. A business has to be sure that it has a valid tax invoice and that it is actually entitled to reclaim the input tax. For example, hotel accommodation purchased by an employee for the purposes of the business will qualify for input tax deduction (where the employee keeps the tax invoice) but this would not normally be the case for a business using a self-employed contractor. In the latter case the supply would be to the contractor not the business and therefore the business would not normally be entitled to reclaim input tax.
An expense is a cost component of the service a business supplies. Payments made on behalf of the client and recharged and where the supply is to the client are called disbursements and no VAT is chargeable. These are often statutory charges due by the client but paid on their behalf as part of the service. The key point is that the supply for VAT purposes is not to the business but to the client whereas for an expense the supply is to the business. There are a number of rules in relation to disbursements and a business has the option of treating a disbursement as an expense but not vice versa. It is usually only an advantage to treat a payment as a disbursement if the supplier did not charge VAT on it or if the client cannot reclaim the VAT.
If in doubt charge VAT on the full value of the expenses!