t = Table[Graphics[Circle[RandomReal[1, 2], RandomReal[1, 2]]], 20];
We can add a button in Manipulate
to paste the current content in the next cell:
pasteButton = Button[Pane["Paste", ImageSize -> {250, Automatic},
Alignment -> Center],
SelectionMove[EvaluationNotebook[], Next, Cell];
NotebookWrite[EvaluationNotebook[], ToBoxes@#];
SelectionMove[EvaluationNotebook[], Previous, Cell];
SelectionMove[EvaluationNotebook[], After, CellContents]] &;
Manipulate[t[[i]],
{{i, 1}, Range[20], Animator, AnimationRunning -> False,
ImageSize -> 250,
AppearanceElements ->
{"ProgressSlider", "StepLeftButton", "StepRightButton",
"PlayPauseButton", "DirectionButton"}},
Dynamic @ pasteButton[t[[i]]],
ContentSize -> {250, 200}, Alignment -> Center]

Update: Organize buttons in a row:
pasteButton2 =
Button[Style["Paste", 10],
SelectionMove[EvaluationNotebook[], Next, Cell];
NotebookWrite[EvaluationNotebook[], ToBoxes@#];
SelectionMove[EvaluationNotebook[], Previous, Cell];
SelectionMove[EvaluationNotebook[], After, CellContents]] &;
Manipulate[t[[i]],
Row[{Control @ {{i, 1}, Range[20], Animator, AnimationRunning -> False,
ImageSize -> 250,
AppearanceElements -> {"ProgressSlider", "FasterSlowerButtons",
"PlayPauseButton", "DirectionButton"}},
Dynamic @ pasteButton2[t[[i]]]}], ContentSize -> {300, 200},
Alignment -> Center]
