That is correct, there are a number of unwanted, or tramp, metals (Cu, Sn, Sb, As) that enter the recycling stream from, for example, car bodies that are ground into scrap without removing all the copper wiring, or tin-coated steel cans. Antimony and arsenic tend to creep in from low-quality and low-cost primary iron sources.
The answer to the question is no. Recycled steel is mixed as evenly as possible from varied sources, its composition is measured, and then pure iron is added as needed to dilute the tramp metals to tolerable levels for resale or further processing, such as meeting a specific steel grade for a specific product or application. Stainless steels and other high-alloy grades which are known at recycling time are processed separately due to the value of Ni, Cr, etc.
It is currently uneconomical to reprocess iron to remove tramp elements, and so it simply isn't done at all. Two books mention the process as a regular and economical one: (Minerals, Metals and Sustainability: Meeting Future Material Needs, p. 284, starting at "dilution") and (Steel Production: Processes, Products, and Residuals, starting on p. 104, read until it isn't relevant anymore). The reason it is uneconomical is that the tramp elements react more weakly with oxygen than iron at constant temperature, so to remove them by oxidation would require oxidizing all of the iron first. The reason for this is thermodynamic, and predicated on the fact that among competing reactions, those with the largest decreases in free energy proceed virtually to completion prior to other reactions even starting, especially with large differences in free energy among the competing reactions. To determine which reactions have the largest decreases, an Ellingham diagram may be used.
In the Ellingham diagram below, the horizontal axis is temperature, the vertical axis is change in Gibbs free energy. The lines running across the diagram at various angles correspond to free energy change caused by element oxidation reactions with oxygen, as a function of temperature. In our case, the diagram may be read by choosing a temperature of interest, and reading up from the bottom to find the first element to react with oxygen. For example, if we have steel with Fe, Mn, Sn, and Cu in it, we can see that at 1000K then Mn, Fe (to FeO), Sn and Cu are the order of largest to smallest drop in free energy.
Granted, the temperature of interest is closer to 1900K (above the melting point of iron), but the general trends of each Gibbs free energy change function continue to the right on the diagram and iron remains below the tramp elements Cu, Sn, As and Sb at practical temperatures, and likely to their respective boiling points. As a result, removing tramps from Fe would require oxidizing effectively all of the iron first. And because Sn, Sb, As and Cu are slightly soluble in iron, they require separation via chemical reaction.
One can see the solubility of tramps from their phase diagrams with iron, of which I have posted Sb-Fe below. The diagram has temperature against composition, with each contiguous 2D region composed of either one phase, or a mixture of the two phases to its left and right, which are in equilibrium at that combination of temperature and composition. At the bottom left we see that for small amounts of Sb and room temperature, there is a contiguous region which in this case denotes a single phase, or alpha-Fe (the kind we are familiar with). Because there is Sb present, and it is in a single phase, it must be dissolved in the iron. The same is true, with varying severity, of the other tramps.
As Chris H commented, there is a question also of when other alloying elements are controlled. Generally alloy addition is controlled as close to solidification as possible, to minimize alloy loss.
Scrap is melted in bulk in an electric arc furnace. If the scrap stream is sufficiently mixed, then the tramp concentration may be estimated based on past usage and the primary iron is added prior to chemistry analysis to compensate for the estimate. The bulk is then melted, oxygen is removed via the addition of elements at the bottom of the Ellingham diagram, specifically Ca and Al, and the molten metal is transferred to one or more highly insulated ladles. The Ca and Al rapidly react with oxygen dissolved in the melt to create low-density oxide slag which floats and is removed mechanically. Chemistry is taken after this process, and if the tramps are sufficiently diluted, the metal is transferred to ladles. If not, sufficient primary iron is added to dilute the melt.
Once in the ladle, additional alloying elements are added. They are not added earlier because of the Ellingham diagram: most alloying elements including Mn, Mo, Cr, V, C, etc. have greater free energy loss than Fe, and so react first. In other words, they fade. To avoid expensive alloy addition fading, they are added as close to the solidification process as possible. Additionally, by removing oxygen using Al and Ca first, there is less oxygen dissolved in the iron to react with the more expensive alloying elements. Once in the ladle, there is very little liquid-atmosphere interface turbulence, so the diffusion of new oxygen into the liquid iron is relatively slow. There is of course still a time limit, and holding a ladle for too long will cause alloy fading. After alloy addition, chemistry is checked, and then the ladle is poured.
Edited to add sources.
Edited to add discussion of alloy control.