Normally no. Under a lease, the lender remains the owner of the asset, so the lender claims the capital allowances — not you. Instead, your relief usually comes through deducting the rentals as a business expense, spread across the term in line with how the lease is treated in your accounts.
This is one of the clearest practical differences between leasing and hire purchase. On HP you own the asset for tax purposes and claim allowances; on a lease you do not, and you expense the rentals instead. Neither is automatically better — it depends on whether your business can actually use allowances in the year of purchase, and on your wider profit position.
There is a further complication. Accounting standards may require a finance lease to be recorded on your balance sheet with a matching liability, while the tax treatment can follow a different route. The deduction you get is not always simply the cash you paid in the year, and which accounting framework your accounts are prepared under affects how the numbers fall out.
Long funding leases are a specific exception where the lessee, not the lessor, may claim allowances. The conditions are technical and turn on the term of the lease relative to the life of the asset. If you think a lease might fall into that category, that is a question for your accountant before signing.
This is general information, not tax advice. Confirm your own position with your accountant or HMRC.