Oh, and your random number in api4 is not actually defined by your api function (you can verify by evaluating CloudImport on api4). Notice that the values of 'a' are the same as previous evaluations of api3. Kernel memory is not cleared immediately! If you wait awhile, you'll find that a is no longer defined when you evaluate api4.

I Changed your package to +1 in {x+1,a} as below

BeginPackage["qpackage`"]
Unprotect[rFunc];

rFunc::usage="does it keep same random number?"

Begin["`Private`"]
a=RandomReal[];
rFunc[x_] := {x+1,a}

End[]
EndPackage[]

api3 with load of package and api4 without loading the package

api3 = CloudDeploy[APIFunction[{"x" -> "Number"},
   (Get["qpackage`"];
     qpackage`rFunc[#x])
    &],
  CloudObject["api3"],
  Permissions -> "Public"]


api4 = CloudDeploy[APIFunction[{"x" -> "Number"},
   (qpackage`rFunc[#x])
    &],
  CloudObject["api4"],
  Permissions -> "Public"]

api3 delivers 1 added to 4 and 8 unique numbers

Table[URLExecute[api3, {"x" -> 4}], {8}] // Tally

{{{5, 0.977889}, 1}, {{5, 0.672346}, 1}, {{5, 0.600011}, 
  1}, {{5, 0.949759}, 1}, {{5, 0.811612}, 1}, {{5, 0.625906}, 
  1}, {{5, 0.530472}, 1}, {{5, 0.658866}, 1}

}

api4 delivers 1 added to each i but no unique random numbers anymore. They repeat every 4 times. So the function is active and can be used if defined delayed?

Table[URLExecute[api4, {"x" -> i}], {i, 1, 8}] 

{{2, 0.530472}, {3, 0.658866}, {4, 0.811612}, {5, 0.625906}, {6, 
  0.530472}, {7, 0.658866}, {8, 0.811612}, {9, 0.625906}}

One other test to see what is happening: edit the package file that is saved in the cloud. I changed the definition of rFunc to return {#x,a,1}. Then I evaluated the api:

In[6]:= URLExecute[CloudObject["api"], {"x" -> 3}]
Out[6]= {3, 0.876432, 1}

So the package definitions are clearly not saved in the deployed function.

I think this shows that the package is loaded every evaluation in the cloud: define a random number in the package, 'a'.

My package:

BeginPackage["qpackage`"]
Unprotect[rFunc];

rFunc::usage="does it keep same random number?"

Begin["`Private`"]
a=RandomReal[];
rFunc[x_] := {x,a}

End[]
EndPackage[]

Now deploy the function and evaluate as you have done:

CopyFile["qpackage.wl", CloudObject["qpackage.wl"]];
api = CloudDeploy[
  APIFunction[{"x" -> "Number"}, (CloudGet[CloudObject["qpackage.wl"]];
     qpackage`rFunc[#x]) &], CloudObject["api"], Permissions -> "Public"];

And evaluate a bunch of times

In[22]:= Table[URLExecute[api, {"x" -> 3}], {4}]
Out[22]= {{3, 0.425269}, {3, 0.898631}, {3, 0.808372}, {3, 0.146791}}

Note that the value of 'a' changes with each evaluation. If I do the same with Mathematica on my desktop, the value of 'a' is fixed.

In[2]:= << qpackage`
In[3]:= rFunc /@ ConstantArray[3, 4]
Out[3]= {{3, 0.495344}, {3, 0.495344}, {3, 0.495344}, {3, 0.495344}}

I cannot reproduce the loading of the package into the ContextPath on your local machine. Perhaps you evaluated the APIFunction locally before deploying? If not, please let me know the $Version you are using, and I'll investigate further.