Your bank shouldn't be charging you money for depositing cash into your account.
To answer the tax question, from my understanding you need to declare any income (that isn't already declared by your employer). If that money doesn't come from an income (for example, you're using your savings or your family is helping you with money), you shouldn't need to declare it or pay tax from it. If you are making money from online freelancing and getting paid into your Moldovan bank account and then optionally moving money into your UK bank, you would likely need to declare this income as you are a tax resident in the UK and need to pay tax from any earnings you make. But the act of transferring money around on its own isn't considered an income.
Obviously, you may attract some kind of attention from the transfer companies or the banks if you start moving huge sums of money around (due to the money laundering laws and hence financial companies and banks checking where the money comes from, which will 99.99% not happen to you if you are moving £1000 or less in a month). Either way Personal Finance & Money Stack Exchange is definitely a better place to answer this part of the question more appropriately.
You should probably also consider other cheap ways to transfer money from abroad into the UK: This can actually save you money and hassle with handling cash.
1. I recommend opening up a Revolut account.
You can then top it up with GBP using your bank card (can be either debit or credit card) from your Moldovan bank, and after that transfer it to your main UK bank. There are no fees and no fixed limits (obviously if you're topping up thousands of pounds daily you might raise eyebrows). The only fee you might get is foreign currency charge by your Moldovan bank for making a payment in British pounds. Some banks charge around 3% for that, some charge 1%, some charge nothing.
P.S. Normally, I would suggest topping up Revolut in the original currency of your bank to avoid paying fees to your bank, and then doing a free exchange into pounds within Revolut if done on Monday-Friday (free for up to £5000 a month). But you can't have a Moldovan leu account in Revolut, so this doesn't apply to you and should use the above option of directly topping up in GBP.
2. A not-as-simple-as-Revolut option but can work for some people if they don't mind bunch of extra steps, I used to do this myself personally. If you need a maximum of £500-£1000 per month, you can open a Monzo Bank account.
In the app, go to Payments -> Request -> Share a link. You should get a monzo.me link. This is meant for friends to send you money using a debit card (to pay for stuff easily). But you can use it yourself to send money to yourself from a Moldovan debit card (I haven't tried it with Moldovan card myself but it has worked for me with a foreign non-UK debit card).
There is a limit of £500 per month per debit card used for the payment, £100 limit per single transaction, and an overall limit of £1000 for top ups using monzo.me into your account. Therefore, if you want to top up £500 you will need one Moldovan debit card and make 5x £100 monzo.me payments. If you want £1000 = 2x Moldovan debit cards and 10x monzo.me payments.
If your bank doesn't charge fees for payments in foreign currency (in this case GBP), this option is completely free.
3. Otherwise, TransferWise is a very simple option and is normally much cheaper than making an international transfer from your bank account.
Also, there are alternatives to TransferWise that work in a similar fashion and in some cases might be even cheaper than TransferWise. I have personally used XendPay though it may not work for you as this particular one is limited to USD/EUR/GBP as your initial currency. There was definitely at least one more more similar website but cannot remember the name, I would normally check on each of them how much would I get from my transfer and then used the one that was the cheapest (gave me the most of money after the transfer).
If your current bank doesn't offer free payments in a foreign currency, you might be able to find another bank in Moldova that offers a cheaper fee or is completely free for this.
Also, different banks charge different fees for ATM withdrawals abroad. It might be possible that there is a bank in Moldova that can has cheaper ATM withdraw fees than your current one. I remember, in my EU country, most banks charged 3% for ATM withdrawals abroad but there was one bank that had a flat fee of 1 euro per ATM withdrawal and no foreign currency fees.