I have a gap about 1/2 inch or a little less under my prehung door sill. when you step on it flex a little. Could I use concrete patch mix in a tube and squeeze it under the door sill to fill the gap?

[color=#141414]We own a home with a 5' x 7' pad at the threshold to the front door. Leading up to it is a narrow concrete path about 18" wide. Over time a large root from a nearby tree has compromised the pad and path by growing directly under the point where the pad and path meet, lifting both. The result is that when it rains the water pools towards the door, and during a typical central Florida afternoon downpour the door and framework get a lot of splash - it got so bad that the framework rotted badly and I had to replace the entire frame.[/color] [color=#141414]We have tried and tried to find someone to break up and replace the pad and small section of the compromised path, but after weeks of leaving messages and sending out emails had to give up - it was either too small a job for pro concrete companies, or we couldn't find any reputable handymen willing to do the job - I literally wrote to dozens of people and received next to no replies. The one person who responded gave us the runaround for two weeks just getting to the house until we finally gave up. So basically it's down to me doing the job. I can do almost anything within the house from top to bottom, but working with concrete at this level is not something I've ever done other than setting fence posts in place. I have worked on concrete gangs at major constructions sites like dams, but that's an entirely different kettle of fish![/color] [color=#141414]I've watched a slew of clips on how to do it, and I think I can manage it, but I have a couple of points I need clarification on. Firstly, some people seem to suggest that I don't need to put in mesh or rebar for a 4" pad that's just being used for a threshold - what's the opinion on that? I'm happy either way - I just want to do the job properly. The biggest concern though is rain. It's monsoon season in central Florida, which means that most afternoons we typically get a major downpour for a few hours. I've been waiting for a window of opportunity for the last couple of weeks, but the forecasts have been pretty abysmal. This week is a perfect example, with more than 50% chance of thunderstorm activity through to Wednesday, then 30% chance of rain activity for Thursday and Friday. I'm particularity concerned about it because the area in question is in a corner of the roof overhang, and even though there's a gutter in place it does get some leaks, and in a really heavy downpour the water is too much to handle and it turns into a waterfall. I thought I could perhaps cover it all with plastic, but I read that plastic can adhere to cement, plus I'm wondering if the water runoff could seep back underneath and compromise the concrete. Is there a way round this, or do I simply wait it out until I get a suitable window of opportunity with the weather?[/color]