It could be low on refrigerant. When was the last time it a preventative maintenance check by a qualified HVAC tech?

Heat pumps take a while to get going. It could be low on refrigerant / it could be normal operation.

The colder it is outside the less heat there is in the outdoor air to bring inside. (Heat pumps do not 'create heat' --- they extract heat from cold air and then move it from outside to inside.) The colder it is outside the more challenging this is.

Anything below 98 degrees will feel cold to the human body.

Emergency heat: if the heat pump enters defrost mode, you will need a secondary form of heat. Sometimes this is the more costly electric strip heat, could be natural gas, propane etc.

May actually not be working at all and just blowing cold air until the aux heat kicks in. How long does it take to start blowing warm air?

just guessing...a service call from a heat pump pro is indicated...

May actually not be working at all and just blowing cold air until the aux heat kicks in. How long does it take to start blowing warm air?

Mine takes 5 min or so to start warming up and it's an inverter heat pump and I am in a mostly hot sticky climate. I keep it chilly in the house so those are other factors to contend with. (The inverter starts slow, then ramps up)

I think a single speed heat pump would ramp up quicker and I doubt it would take more than a couple of minutes, depending on outdoor air temps and how cold it is inside. (single speed is either on or off, so you get the full output capacity within a few minutes of it coming on.) ---obviously the cost to operate one of these single speed HP's costs more than the inverter.

Given that he shuts it off for a minute and turns it back on leads me to believe it's probably normal operation.

The OP doesn't clarify if he's new to heat pumps.

How long exactly is "eventually get warmer?"

And what is the airflow temperature at an output register (near to the air handler's location) when it does get warm?

Turning it off for (literally) one minute (or two mins or three mins, etc.) and back on, the compressor probably won't turn back on for ~5 minutes due to pressure equalization protection ... so there's part of the delay in regaining heat output.

Just an anecdotal little story with no facts...details on system-brand, model, size, and age. Location? .OP should request a service call from an experienced Heat pump pro..BTW, when was last time HP system was serviced?

IMO

This is happening when it is in cooling mode. It seems to operate normally(blowing cold air) but then the air gradually becomes warmer. Then if I turn it off for a moment and back on it will be cold again. The technician came out and added 1 lbs of refridgerant in january. It also says that 2lbs were added in august of 2018 so there is obviously a leak.

I am worried about having to replace the entire unit since the technician told me he couldn't fix the leak. He said the leak was probably in the coil and that it couldnt be repaired. Does this sound like a probable situation or is this guy just trying to sell me a new unit? If the leak is so bad that it needs to be replaced why does it work sometimes?

right now it is blowing 46 degree air and it is 52 degrees outside. Shouldn't it be like 15 degrees colder than the outside air? I'll try running it when it is hotter outside and measure it then also.

"Shouldn't it be like 15 degrees colder than the outside air?"

No. It doesn't pull outdoor air into the house. Temperature drop (or rise) is based on the INDOOR air temperature, at the return air grill.