Can we charge tenant for advertise fee by changing their mind ?

The rent is $40 below market value, since tenant is good, so notify the PM to increase $20 after the lease expired, tenant has sent a letter to inform that they will move on when the lease expired. Fair enough, so PM has put the IP back on market and advertise with the market value.

There are still 1 more week before the lease expired, the tenant has changed their mind, and would like to stay on with the initial of $20 increase, I don't mind the tenant to stay on, but since the PM has already advertise, and has conducted some open inspection, I am pretty sure the PM would charge me for those fees.

I would like to know should I bear those fees or I can get them back from the tenant...?

i'm going to bet you advertised the property at market value (ie, $40 above what you charge now).
Therefore, I would tell the tennant he can stay on for market rent of $40/wk above what he pays now.... or maybe $30 - citing that you had a number of applications at the full market rent of $40 more than he is paying now.

Your tennant cant have the best of both worlds :)
Let the current tenants know they can have the property at the market value which you are advertising it for.

Yep, agreed.

You've given them massively reduced rent and you told them they could stay for a $20 increase - which they though (i'm presuming) was you being a greedy LL and the tenant has stuck up their nose and thought they could find something comparable at the same price and found they couldn't and came running back. This is why you say, too bad so sad, they had the opportunity, they knocked it back - they can suck it up.
Why on earth did you let it get $40 behind market in the first place. Good tenant or not, they should always be on market rates.
The rent is $40 below market value, since tenant is good

Sorry, this doesn't sit well with me. You're in the business of property, not charity.

As Skater said: tenants have to pay market rent. End of discussion.

What has happened, I bet... is that the tenant has now done their research and has discovered, lo and behold, that they'll be paying $20 *more* per week to rent ANY other similar property in the area. That's over $1000 a year. Plus moving costs. Plus reconnecting services, changing address everywhere, redirecting mail and so on.

In answer to your question, you'll probably have to bear those fees. Or maybe your property manager is nice enough to waive them. Ask. Consider it a cost of learning - always charge market rent! :D
What price is there for peace of mind? You said they were a good tenant.....

Oh I know the overwhelming advice is to enforce your 'landlordial powers' and increase the rent to full value immediately and teach the tenant the value of your generosity.

Then they may enforce their 'vote with their feet powers' and find somewhere with that same 'market rent' you would enforce. Maybe they are happy to do that too. Despite the cost.

But if it really is about the $$$$$$$$ then the extra $20 per week loss could be good value for you anyway.

Consider as you say the advertising costs and the one week or so of vacancy (if your lucky) and the one weeks letting fee for the agent, add them up and work out how many weeks it will take you to recoup that amount at the higher rate. At $160 a week it will take 4 months, at $400 a week it will take near 10 months. Then your even.

$40 a week jump in rent is hard to find for most people. I can see why they would resist.

I would relet at the $20 increase for 3 or 6 months and advise that it will go up the other $20 at the end.

Positve or negative geared - you have done without the $40 for some time, is it really worth the hassle if they are "good tenants"?
I think GeeJones talks a lot of sense. I try to have "new" tenants always go in at market rates, but if, over a couple of years it ends up that they are $10 or $20 under what I could get if I started with fresh tenants, it does not overly worry me, especially as pinpointing "market rent" in an older house is not an easy science.

I know that most tenants don't stay for years and then it is catch up time.

A lot of the time, it depends on what the tenants are like. I would much rather have great tenants and accept $10 per week less than risk having an empty house with any little maintenance things that I would have to address before a new tenant can go in.

I don't think that makes me a "soft landlord" but a landlord who looks at how much it will cost to have a house sit empty for a couple of weeks.

We have had an agent find a tenant the last four times, and whilst they have found quality tenants (just as we could have done) there is just not the same incentive to do it quickly. We have just accepted new tenants and when they move in, it will be four weeks since the old tenants moved out.

It is not a financial issue for us, but ten years ago, that money would have meant a struggle for us to meet our bills.

I do have a relationship with our tenants as we self-manage, but that relationship does not extend to me being a doormat. Rather, I think I am a sensible landlord and weigh up how far I can raise the rent without pushing the tenant to look elsewhere to see what they can get for the money. This is particularly more an issue as our houses are old, but in good condition, and for the same money in our area a tenant could rent a lovely townhouse with two bathrooms and better facilities.
All my new tenants are also started off with the market rent, but throughout the years, IF the tenants are trouble free, I would loosing the "increase" a bit each year, until the tenant is moving out, then I would advertise with the market rent for the new tenant again.

I know investment is not charity, but that is my strategy to filter the good tenant for the IP, to reduce vacancy, to reduce the hassle and maintainence down to a minimum, or you can say that is the rewards for being a good tenant...:D

To be fair with the tenant, according to the PM, they choose to move out is due to the uncertain of their contract with the company at the time, but they have just received a notification from the company that their contract is extended.

Have chatted with the PM, he agrees to charge me $55 for the advertising cost only and waive the rest if the existing tenant stay on, so I agree with GeeJones's method to have another rent review in 6 months time.

But with that $55, well, simply tell my kids no Mac this week...:p