How to add a circle on the top right corner of a button above the background and behind the border?

Question

I wanna achieve the following result by using CSS:

So basically I want the circle to be on top of the button background but behind its border, with the butt

Answer 1

One idea is to integrate the missing borders inside the circle

.container {
  margin-top: 30px;
}

button {
  font-size: 20px;
  border: 2px solid black;
  padding: 8px 20px;
  position: relative;
}

button:before {
  content: "";
  position: absolute;
  top: -1px;
  right: -1px;
  transform:translate(50%,-50%);
  width: 40px;
  height: 40px;
  border-radius: 50%;
  background: 
   linear-gradient(black,black) left  /50% 2px,
   linear-gradient(black,black) bottom/2px 50%,
   #4da6ff;
 background-repeat:no-repeat;
}

<div class="container">
  <button>Test Button</button>
</div>

Or you can simply consider mix-blend-mode. You have to pay attention to the value used as it will depend on the combination of the colors. In this case, the suitable one is darken

.container {
  margin-top: 30px;
}

button {
  font-size: 20px;
  border: 2px solid black;
  padding: 8px 20px;
  position: relative;
}

button:before {
  content: "";
  position: absolute;
  top: -1px;
  right: -1px;
  transform:translate(50%,-50%);
  width: 40px;
  height: 40px;
  border-radius: 50%;
  background: #4da6ff;
  mix-blend-mode:darken;
}

<div class="container">
  <button>Test Button</button>
</div>

A third way more fancy with only backgrounds:

button {
  font-size: 20px;
  border:0 solid transparent;
  border-top-width:24px;
  border-right-width:24px;
  padding: 8px 20px;
  background:
     linear-gradient(black,black) top   /100% 2px,
     linear-gradient(black,black) bottom/100% 2px,
     linear-gradient(black,black) left  /2px 100%,
     linear-gradient(black,black) right /2px 100%,
     radial-gradient(circle, #4da6ff 19px,transparent 20px) left bottom/200% 200% padding-box border-box,
     #e2e2e6 padding-box;
  background-repeat:no-repeat;
}

<div class="container">
  <button>Test Button</button>
</div>

Another idea is to place the circle behind the element and cut the background:

.container {
  margin-top: 30px;
  position:relative;
  z-index:0;
}

button {
  font-size: 20px;
  border: 2px solid black;
  padding: 8px 20px;
  position: relative;
  background:radial-gradient(circle at top right,transparent 19px,#e2e2e6 20px);
}

button:before {
  content: "";
  position: absolute;
  z-index:-1;
  top: -1px;
  right: -1px;
  transform:translate(50%,-50%);
  width: 40px;
  height: 40px;
  border-radius: 50%;
  background:#4da6ff;
}

<div class="container">
  <button>Test Button</button>
</div>

Answer 2

Use a pseudo element (::after) to draw the border above the circle:

.container {
  margin-top: 30px;
}

button {
  position: relative;
  font-size: 20px;
  border: none;
  padding: 8px 20px;
}

button::before {
  position: absolute;
  top: -20px;
  right: -20px;
  width: 40px;
  height: 40px;
  border-radius: 25px;
  background: #4da6ff;
  content: '';
}

button::after {
  position: absolute;
  top: -2px;
  right: -2px;
  bottom: -2px;
  left: -2px;
  border: 2px solid black;
  content: '';
}

<div class="container">
  <button>Test Button</button>
</div>